How does this infinite series $1-frac14+frac17-frac110+cdots$ simplify to an integral $int_0^1fracdx1+x^3$? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)When can a sum and integral be interchanged?Infinite Series $1+frac12-frac23+frac14+frac15-frac26+cdots$does the infinite series $sum^infty_n=1 (-1)^n frac log(n)n$ converge?Simplify Infinite Series Involving Gamma Function $Gamma$How to simplify this integral?Sum of an infinite series $(1 - frac 12) + (frac 12 - frac 13) + cdots$ - not geometric series?How to find the value of this integral: $int_0^nfrac 1x dx$Does this integral $int_0^infty fracdx(1+e^x)(a+x)$ have a closed form?Does this infinite series converge or diverge?Value of this convergent series: $frac13!+frac25!+frac37!+frac49!+cdots$Calculating a series for$int_0^1 fracx^p-11+x^q dx$ by interchanging sum with integral

Should I use a zero-interest credit card for a large one-time purchase?

An adverb for when you're not exaggerating

Is CEO the profession with the most psychopaths?

Why are the trig functions versine, haversine, exsecant, etc, rarely used in modern mathematics?

Has negative voting ever been officially implemented in elections, or seriously proposed, or even studied?

How to react to hostile behavior from a senior developer?

What is the longest distance a player character can jump in one leap?

If my PI received research grants from a company to be able to pay my postdoc salary, did I have a potential conflict interest too?

What's the meaning of "fortified infraction restraint"?

Can you use the Shield Master feat to shove someone before you make an attack by using a Readied action?

Why do we bend a book to keep it straight?

If a VARCHAR(MAX) column is included in an index, is the entire value always stored in the index page(s)?

Can a new player join a group only when a new campaign starts?

Fundamental Solution of the Pell Equation

Significance of Cersei's obsession with elephants?

Why are both D and D# fitting into my E minor key?

Amount of permutations on an NxNxN Rubik's Cube

Do I really need to have a message in a novel to appeal to readers?

Crossing US/Canada Border for less than 24 hours

Where are Serre’s lectures at Collège de France to be found?

Why aren't air breathing engines used as small first stages

How to Make a Beautiful Stacked 3D Plot

What do you call the main part of a joke?

Is it cost-effective to upgrade an old-ish Giant Escape R3 commuter bike with entry-level branded parts (wheels, drivetrain)?



How does this infinite series $1-frac14+frac17-frac110+cdots$ simplify to an integral $int_0^1fracdx1+x^3$?



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)When can a sum and integral be interchanged?Infinite Series $1+frac12-frac23+frac14+frac15-frac26+cdots$does the infinite series $sum^infty_n=1 (-1)^n frac log(n)n$ converge?Simplify Infinite Series Involving Gamma Function $Gamma$How to simplify this integral?Sum of an infinite series $(1 - frac 12) + (frac 12 - frac 13) + cdots$ - not geometric series?How to find the value of this integral: $int_0^nfrac 1x dx$Does this integral $int_0^infty fracdx(1+e^x)(a+x)$ have a closed form?Does this infinite series converge or diverge?Value of this convergent series: $frac13!+frac25!+frac37!+frac49!+cdots$Calculating a series for$int_0^1 fracx^p-11+x^q dx$ by interchanging sum with integral










2












$begingroup$


How does the infinite series below simplify to that integral?



$$1-frac14+frac17-frac110+cdots=int_0^1fracdx1+x^3$$



I thought of simplifying the series to the sum to infinity of $frac16n-5 - frac16n-2$, but this did not help.






integration sequences-and-series power-series







share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    en.wikipedia.org/wiki/Binomial_series $$(1+x^3)^-1$$
    $endgroup$
    – lab bhattacharjee
    Apr 13 at 16:05










  • $begingroup$
    It can be obtained by integrating term by term the infinite series expansion of $(1+x^3)^-1$.
    $endgroup$
    – StubbornAtom
    Apr 13 at 16:11















2












$begingroup$


How does the infinite series below simplify to that integral?



$$1-frac14+frac17-frac110+cdots=int_0^1fracdx1+x^3$$



I thought of simplifying the series to the sum to infinity of $frac16n-5 - frac16n-2$, but this did not help.






integration sequences-and-series power-series







share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    en.wikipedia.org/wiki/Binomial_series $$(1+x^3)^-1$$
    $endgroup$
    – lab bhattacharjee
    Apr 13 at 16:05










  • $begingroup$
    It can be obtained by integrating term by term the infinite series expansion of $(1+x^3)^-1$.
    $endgroup$
    – StubbornAtom
    Apr 13 at 16:11













2












2








2


3



$begingroup$


How does the infinite series below simplify to that integral?



$$1-frac14+frac17-frac110+cdots=int_0^1fracdx1+x^3$$



I thought of simplifying the series to the sum to infinity of $frac16n-5 - frac16n-2$, but this did not help.






integration sequences-and-series power-series







share|cite|improve this question











$endgroup$




How does the infinite series below simplify to that integral?



$$1-frac14+frac17-frac110+cdots=int_0^1fracdx1+x^3$$



I thought of simplifying the series to the sum to infinity of $frac16n-5 - frac16n-2$, but this did not help.






integration sequences-and-series power-series




integration sequences-and-series power-series






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 9 hours ago









HAMIDINE SOUMARE

2,570417




2,570417










asked Apr 13 at 16:03









ShreeShree

134




134







  • 1




    $begingroup$
    en.wikipedia.org/wiki/Binomial_series $$(1+x^3)^-1$$
    $endgroup$
    – lab bhattacharjee
    Apr 13 at 16:05










  • $begingroup$
    It can be obtained by integrating term by term the infinite series expansion of $(1+x^3)^-1$.
    $endgroup$
    – StubbornAtom
    Apr 13 at 16:11












  • 1




    $begingroup$
    en.wikipedia.org/wiki/Binomial_series $$(1+x^3)^-1$$
    $endgroup$
    – lab bhattacharjee
    Apr 13 at 16:05










  • $begingroup$
    It can be obtained by integrating term by term the infinite series expansion of $(1+x^3)^-1$.
    $endgroup$
    – StubbornAtom
    Apr 13 at 16:11







1




1




$begingroup$
en.wikipedia.org/wiki/Binomial_series $$(1+x^3)^-1$$
$endgroup$
– lab bhattacharjee
Apr 13 at 16:05




$begingroup$
en.wikipedia.org/wiki/Binomial_series $$(1+x^3)^-1$$
$endgroup$
– lab bhattacharjee
Apr 13 at 16:05












$begingroup$
It can be obtained by integrating term by term the infinite series expansion of $(1+x^3)^-1$.
$endgroup$
– StubbornAtom
Apr 13 at 16:11




$begingroup$
It can be obtained by integrating term by term the infinite series expansion of $(1+x^3)^-1$.
$endgroup$
– StubbornAtom
Apr 13 at 16:11










3 Answers
3






active

oldest

votes


















12












$begingroup$

$$int_0^1fracdx1-(-x)^3=int_0^1sum_n=0^infty(-x)^3ndx=sum_n=0^infty(-1)^3nint_0^1x^3ndx$$
$$=sum_n=0^inftyfrac(-1)^3n3n+1= 1-frac14+frac17-frac110+cdots $$






share|cite|improve this answer











$endgroup$












  • $begingroup$
    For those who aren't clear on the first equality, it's a geometric series formula.
    $endgroup$
    – Alexis Olson
    Apr 13 at 19:35










  • $begingroup$
    Why can we interchange the sum and integration? what is the idea? thanks. @AlexisOlson If you are free, please respond. Thanks.
    $endgroup$
    – StammeringMathematician
    22 hours ago






  • 1




    $begingroup$
    @StammeringMathematician Check out this question.
    $endgroup$
    – Alexis Olson
    14 hours ago










  • $begingroup$
    @AlexisOlson Thanks a lot.
    $endgroup$
    – StammeringMathematician
    12 hours ago


















4












$begingroup$

for $x$ real, $ngeq 0$ integer
beginalignfrac11+x^3&=frac1-(-x^3)^n+11-(-x^3)+frac(-x^3)^n+11-(-x^3)\
&=frac1-(-x^3)^n+11-(-x^3)+frac(-x^3)^n+11+x^3\
endalign



For $xneq 1$, $ngeq 0$ integer, beginalignsum_k=0^n x^k=frac1-x^n+11-xendalign



Therefore,
beginalignint_0^1 frac11+x^3,dx&=int_0^1 left(sum_k=0^n (-x^3)^kright),dx+int_0^1 frac(-x^3)^n+11+x^3,dx\
&=sum_k=0^n left(int_0^1 (-x^3)^k,dxright)+int_0^1 frac(-x^3)^n+11+x^3,dx\
&=sum_k=0^n frac(-1)^k3k+1+int_0^1 frac(-x^3)^n+11+x^3,dx\
endalign



For $xin[0;1],ngeq 0$, integer,
beginalignfracx^3(n+1)1+x^3leq x^3(n+1)endalign
and,
beginalignint_0^1 x^3(n+1),dx=frac13n+4endalign
Therefore,
beginalignleft|int_0^1 frac(-x^3)^n+11+x^3,dxright|leq frac13n+4endalign
beginalignleft|int_0^1 frac11+x^3,dx-sum_k=0^n frac(-1)^k3k+1right|leq frac13n+4endalign
Therefore,
beginalignboxedint_0^1 frac11+x^3,dx=sum_k=0^infty frac(-1)^k3k+1endalign






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Why can we interchange $sum$ and $int$
    $endgroup$
    – StammeringMathematician
    22 hours ago










  • $begingroup$
    Only finite sums (that is, not series) are used. You can interchange sum and integral signs when the sum involves finite number of terms.
    $endgroup$
    – FDP
    21 hours ago










  • $begingroup$
    Thanks, I got it. Actually I was confused with the answer above yours one as there summation is over infinite terms.
    $endgroup$
    – StammeringMathematician
    21 hours ago










  • $begingroup$
    My computation shows that series converges slowly.
    $endgroup$
    – FDP
    20 hours ago


















2












$begingroup$

If $lvert xrvert<1$, let$$f(x)=sum_n=0^inftyfracx^3n+13n+1.$$Then $$f'(x)=sum_n=0^infty x^3n=frac11-x^3.$$Thereforebeginalign1-frac14+frac17-frac110+cdots&=lim_xto1f(x)\&=int_0^1f'(x),mathrm dx\&=int_0^1frac11-x^3,mathrm dx.endalign






share|cite|improve this answer









$endgroup$













    Your Answer








    StackExchange.ready(function()
    var channelOptions =
    tags: "".split(" "),
    id: "69"
    ;
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function()
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled)
    StackExchange.using("snippets", function()
    createEditor();
    );

    else
    createEditor();

    );

    function createEditor()
    StackExchange.prepareEditor(
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader:
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    ,
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    );



    );













    draft saved

    draft discarded


















    StackExchange.ready(
    function ()
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3186397%2fhow-does-this-infinite-series-1-frac14-frac17-frac110-cdots-si%23new-answer', 'question_page');

    );

    Post as a guest















    Required, but never shown

























    3 Answers
    3






    active

    oldest

    votes








    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    12












    $begingroup$

    $$int_0^1fracdx1-(-x)^3=int_0^1sum_n=0^infty(-x)^3ndx=sum_n=0^infty(-1)^3nint_0^1x^3ndx$$
    $$=sum_n=0^inftyfrac(-1)^3n3n+1= 1-frac14+frac17-frac110+cdots $$






    share|cite|improve this answer











    $endgroup$












    • $begingroup$
      For those who aren't clear on the first equality, it's a geometric series formula.
      $endgroup$
      – Alexis Olson
      Apr 13 at 19:35










    • $begingroup$
      Why can we interchange the sum and integration? what is the idea? thanks. @AlexisOlson If you are free, please respond. Thanks.
      $endgroup$
      – StammeringMathematician
      22 hours ago






    • 1




      $begingroup$
      @StammeringMathematician Check out this question.
      $endgroup$
      – Alexis Olson
      14 hours ago










    • $begingroup$
      @AlexisOlson Thanks a lot.
      $endgroup$
      – StammeringMathematician
      12 hours ago















    12












    $begingroup$

    $$int_0^1fracdx1-(-x)^3=int_0^1sum_n=0^infty(-x)^3ndx=sum_n=0^infty(-1)^3nint_0^1x^3ndx$$
    $$=sum_n=0^inftyfrac(-1)^3n3n+1= 1-frac14+frac17-frac110+cdots $$






    share|cite|improve this answer











    $endgroup$












    • $begingroup$
      For those who aren't clear on the first equality, it's a geometric series formula.
      $endgroup$
      – Alexis Olson
      Apr 13 at 19:35










    • $begingroup$
      Why can we interchange the sum and integration? what is the idea? thanks. @AlexisOlson If you are free, please respond. Thanks.
      $endgroup$
      – StammeringMathematician
      22 hours ago






    • 1




      $begingroup$
      @StammeringMathematician Check out this question.
      $endgroup$
      – Alexis Olson
      14 hours ago










    • $begingroup$
      @AlexisOlson Thanks a lot.
      $endgroup$
      – StammeringMathematician
      12 hours ago













    12












    12








    12





    $begingroup$

    $$int_0^1fracdx1-(-x)^3=int_0^1sum_n=0^infty(-x)^3ndx=sum_n=0^infty(-1)^3nint_0^1x^3ndx$$
    $$=sum_n=0^inftyfrac(-1)^3n3n+1= 1-frac14+frac17-frac110+cdots $$






    share|cite|improve this answer











    $endgroup$



    $$int_0^1fracdx1-(-x)^3=int_0^1sum_n=0^infty(-x)^3ndx=sum_n=0^infty(-1)^3nint_0^1x^3ndx$$
    $$=sum_n=0^inftyfrac(-1)^3n3n+1= 1-frac14+frac17-frac110+cdots $$







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited yesterday

























    answered Apr 13 at 16:15









    HAMIDINE SOUMAREHAMIDINE SOUMARE

    2,570417




    2,570417











    • $begingroup$
      For those who aren't clear on the first equality, it's a geometric series formula.
      $endgroup$
      – Alexis Olson
      Apr 13 at 19:35










    • $begingroup$
      Why can we interchange the sum and integration? what is the idea? thanks. @AlexisOlson If you are free, please respond. Thanks.
      $endgroup$
      – StammeringMathematician
      22 hours ago






    • 1




      $begingroup$
      @StammeringMathematician Check out this question.
      $endgroup$
      – Alexis Olson
      14 hours ago










    • $begingroup$
      @AlexisOlson Thanks a lot.
      $endgroup$
      – StammeringMathematician
      12 hours ago
















    • $begingroup$
      For those who aren't clear on the first equality, it's a geometric series formula.
      $endgroup$
      – Alexis Olson
      Apr 13 at 19:35










    • $begingroup$
      Why can we interchange the sum and integration? what is the idea? thanks. @AlexisOlson If you are free, please respond. Thanks.
      $endgroup$
      – StammeringMathematician
      22 hours ago






    • 1




      $begingroup$
      @StammeringMathematician Check out this question.
      $endgroup$
      – Alexis Olson
      14 hours ago










    • $begingroup$
      @AlexisOlson Thanks a lot.
      $endgroup$
      – StammeringMathematician
      12 hours ago















    $begingroup$
    For those who aren't clear on the first equality, it's a geometric series formula.
    $endgroup$
    – Alexis Olson
    Apr 13 at 19:35




    $begingroup$
    For those who aren't clear on the first equality, it's a geometric series formula.
    $endgroup$
    – Alexis Olson
    Apr 13 at 19:35












    $begingroup$
    Why can we interchange the sum and integration? what is the idea? thanks. @AlexisOlson If you are free, please respond. Thanks.
    $endgroup$
    – StammeringMathematician
    22 hours ago




    $begingroup$
    Why can we interchange the sum and integration? what is the idea? thanks. @AlexisOlson If you are free, please respond. Thanks.
    $endgroup$
    – StammeringMathematician
    22 hours ago




    1




    1




    $begingroup$
    @StammeringMathematician Check out this question.
    $endgroup$
    – Alexis Olson
    14 hours ago




    $begingroup$
    @StammeringMathematician Check out this question.
    $endgroup$
    – Alexis Olson
    14 hours ago












    $begingroup$
    @AlexisOlson Thanks a lot.
    $endgroup$
    – StammeringMathematician
    12 hours ago




    $begingroup$
    @AlexisOlson Thanks a lot.
    $endgroup$
    – StammeringMathematician
    12 hours ago











    4












    $begingroup$

    for $x$ real, $ngeq 0$ integer
    beginalignfrac11+x^3&=frac1-(-x^3)^n+11-(-x^3)+frac(-x^3)^n+11-(-x^3)\
    &=frac1-(-x^3)^n+11-(-x^3)+frac(-x^3)^n+11+x^3\
    endalign



    For $xneq 1$, $ngeq 0$ integer, beginalignsum_k=0^n x^k=frac1-x^n+11-xendalign



    Therefore,
    beginalignint_0^1 frac11+x^3,dx&=int_0^1 left(sum_k=0^n (-x^3)^kright),dx+int_0^1 frac(-x^3)^n+11+x^3,dx\
    &=sum_k=0^n left(int_0^1 (-x^3)^k,dxright)+int_0^1 frac(-x^3)^n+11+x^3,dx\
    &=sum_k=0^n frac(-1)^k3k+1+int_0^1 frac(-x^3)^n+11+x^3,dx\
    endalign



    For $xin[0;1],ngeq 0$, integer,
    beginalignfracx^3(n+1)1+x^3leq x^3(n+1)endalign
    and,
    beginalignint_0^1 x^3(n+1),dx=frac13n+4endalign
    Therefore,
    beginalignleft|int_0^1 frac(-x^3)^n+11+x^3,dxright|leq frac13n+4endalign
    beginalignleft|int_0^1 frac11+x^3,dx-sum_k=0^n frac(-1)^k3k+1right|leq frac13n+4endalign
    Therefore,
    beginalignboxedint_0^1 frac11+x^3,dx=sum_k=0^infty frac(-1)^k3k+1endalign






    share|cite|improve this answer









    $endgroup$












    • $begingroup$
      Why can we interchange $sum$ and $int$
      $endgroup$
      – StammeringMathematician
      22 hours ago










    • $begingroup$
      Only finite sums (that is, not series) are used. You can interchange sum and integral signs when the sum involves finite number of terms.
      $endgroup$
      – FDP
      21 hours ago










    • $begingroup$
      Thanks, I got it. Actually I was confused with the answer above yours one as there summation is over infinite terms.
      $endgroup$
      – StammeringMathematician
      21 hours ago










    • $begingroup$
      My computation shows that series converges slowly.
      $endgroup$
      – FDP
      20 hours ago















    4












    $begingroup$

    for $x$ real, $ngeq 0$ integer
    beginalignfrac11+x^3&=frac1-(-x^3)^n+11-(-x^3)+frac(-x^3)^n+11-(-x^3)\
    &=frac1-(-x^3)^n+11-(-x^3)+frac(-x^3)^n+11+x^3\
    endalign



    For $xneq 1$, $ngeq 0$ integer, beginalignsum_k=0^n x^k=frac1-x^n+11-xendalign



    Therefore,
    beginalignint_0^1 frac11+x^3,dx&=int_0^1 left(sum_k=0^n (-x^3)^kright),dx+int_0^1 frac(-x^3)^n+11+x^3,dx\
    &=sum_k=0^n left(int_0^1 (-x^3)^k,dxright)+int_0^1 frac(-x^3)^n+11+x^3,dx\
    &=sum_k=0^n frac(-1)^k3k+1+int_0^1 frac(-x^3)^n+11+x^3,dx\
    endalign



    For $xin[0;1],ngeq 0$, integer,
    beginalignfracx^3(n+1)1+x^3leq x^3(n+1)endalign
    and,
    beginalignint_0^1 x^3(n+1),dx=frac13n+4endalign
    Therefore,
    beginalignleft|int_0^1 frac(-x^3)^n+11+x^3,dxright|leq frac13n+4endalign
    beginalignleft|int_0^1 frac11+x^3,dx-sum_k=0^n frac(-1)^k3k+1right|leq frac13n+4endalign
    Therefore,
    beginalignboxedint_0^1 frac11+x^3,dx=sum_k=0^infty frac(-1)^k3k+1endalign






    share|cite|improve this answer









    $endgroup$












    • $begingroup$
      Why can we interchange $sum$ and $int$
      $endgroup$
      – StammeringMathematician
      22 hours ago










    • $begingroup$
      Only finite sums (that is, not series) are used. You can interchange sum and integral signs when the sum involves finite number of terms.
      $endgroup$
      – FDP
      21 hours ago










    • $begingroup$
      Thanks, I got it. Actually I was confused with the answer above yours one as there summation is over infinite terms.
      $endgroup$
      – StammeringMathematician
      21 hours ago










    • $begingroup$
      My computation shows that series converges slowly.
      $endgroup$
      – FDP
      20 hours ago













    4












    4








    4





    $begingroup$

    for $x$ real, $ngeq 0$ integer
    beginalignfrac11+x^3&=frac1-(-x^3)^n+11-(-x^3)+frac(-x^3)^n+11-(-x^3)\
    &=frac1-(-x^3)^n+11-(-x^3)+frac(-x^3)^n+11+x^3\
    endalign



    For $xneq 1$, $ngeq 0$ integer, beginalignsum_k=0^n x^k=frac1-x^n+11-xendalign



    Therefore,
    beginalignint_0^1 frac11+x^3,dx&=int_0^1 left(sum_k=0^n (-x^3)^kright),dx+int_0^1 frac(-x^3)^n+11+x^3,dx\
    &=sum_k=0^n left(int_0^1 (-x^3)^k,dxright)+int_0^1 frac(-x^3)^n+11+x^3,dx\
    &=sum_k=0^n frac(-1)^k3k+1+int_0^1 frac(-x^3)^n+11+x^3,dx\
    endalign



    For $xin[0;1],ngeq 0$, integer,
    beginalignfracx^3(n+1)1+x^3leq x^3(n+1)endalign
    and,
    beginalignint_0^1 x^3(n+1),dx=frac13n+4endalign
    Therefore,
    beginalignleft|int_0^1 frac(-x^3)^n+11+x^3,dxright|leq frac13n+4endalign
    beginalignleft|int_0^1 frac11+x^3,dx-sum_k=0^n frac(-1)^k3k+1right|leq frac13n+4endalign
    Therefore,
    beginalignboxedint_0^1 frac11+x^3,dx=sum_k=0^infty frac(-1)^k3k+1endalign






    share|cite|improve this answer









    $endgroup$



    for $x$ real, $ngeq 0$ integer
    beginalignfrac11+x^3&=frac1-(-x^3)^n+11-(-x^3)+frac(-x^3)^n+11-(-x^3)\
    &=frac1-(-x^3)^n+11-(-x^3)+frac(-x^3)^n+11+x^3\
    endalign



    For $xneq 1$, $ngeq 0$ integer, beginalignsum_k=0^n x^k=frac1-x^n+11-xendalign



    Therefore,
    beginalignint_0^1 frac11+x^3,dx&=int_0^1 left(sum_k=0^n (-x^3)^kright),dx+int_0^1 frac(-x^3)^n+11+x^3,dx\
    &=sum_k=0^n left(int_0^1 (-x^3)^k,dxright)+int_0^1 frac(-x^3)^n+11+x^3,dx\
    &=sum_k=0^n frac(-1)^k3k+1+int_0^1 frac(-x^3)^n+11+x^3,dx\
    endalign



    For $xin[0;1],ngeq 0$, integer,
    beginalignfracx^3(n+1)1+x^3leq x^3(n+1)endalign
    and,
    beginalignint_0^1 x^3(n+1),dx=frac13n+4endalign
    Therefore,
    beginalignleft|int_0^1 frac(-x^3)^n+11+x^3,dxright|leq frac13n+4endalign
    beginalignleft|int_0^1 frac11+x^3,dx-sum_k=0^n frac(-1)^k3k+1right|leq frac13n+4endalign
    Therefore,
    beginalignboxedint_0^1 frac11+x^3,dx=sum_k=0^infty frac(-1)^k3k+1endalign







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Apr 13 at 16:47









    FDPFDP

    6,16211929




    6,16211929











    • $begingroup$
      Why can we interchange $sum$ and $int$
      $endgroup$
      – StammeringMathematician
      22 hours ago










    • $begingroup$
      Only finite sums (that is, not series) are used. You can interchange sum and integral signs when the sum involves finite number of terms.
      $endgroup$
      – FDP
      21 hours ago










    • $begingroup$
      Thanks, I got it. Actually I was confused with the answer above yours one as there summation is over infinite terms.
      $endgroup$
      – StammeringMathematician
      21 hours ago










    • $begingroup$
      My computation shows that series converges slowly.
      $endgroup$
      – FDP
      20 hours ago
















    • $begingroup$
      Why can we interchange $sum$ and $int$
      $endgroup$
      – StammeringMathematician
      22 hours ago










    • $begingroup$
      Only finite sums (that is, not series) are used. You can interchange sum and integral signs when the sum involves finite number of terms.
      $endgroup$
      – FDP
      21 hours ago










    • $begingroup$
      Thanks, I got it. Actually I was confused with the answer above yours one as there summation is over infinite terms.
      $endgroup$
      – StammeringMathematician
      21 hours ago










    • $begingroup$
      My computation shows that series converges slowly.
      $endgroup$
      – FDP
      20 hours ago















    $begingroup$
    Why can we interchange $sum$ and $int$
    $endgroup$
    – StammeringMathematician
    22 hours ago




    $begingroup$
    Why can we interchange $sum$ and $int$
    $endgroup$
    – StammeringMathematician
    22 hours ago












    $begingroup$
    Only finite sums (that is, not series) are used. You can interchange sum and integral signs when the sum involves finite number of terms.
    $endgroup$
    – FDP
    21 hours ago




    $begingroup$
    Only finite sums (that is, not series) are used. You can interchange sum and integral signs when the sum involves finite number of terms.
    $endgroup$
    – FDP
    21 hours ago












    $begingroup$
    Thanks, I got it. Actually I was confused with the answer above yours one as there summation is over infinite terms.
    $endgroup$
    – StammeringMathematician
    21 hours ago




    $begingroup$
    Thanks, I got it. Actually I was confused with the answer above yours one as there summation is over infinite terms.
    $endgroup$
    – StammeringMathematician
    21 hours ago












    $begingroup$
    My computation shows that series converges slowly.
    $endgroup$
    – FDP
    20 hours ago




    $begingroup$
    My computation shows that series converges slowly.
    $endgroup$
    – FDP
    20 hours ago











    2












    $begingroup$

    If $lvert xrvert<1$, let$$f(x)=sum_n=0^inftyfracx^3n+13n+1.$$Then $$f'(x)=sum_n=0^infty x^3n=frac11-x^3.$$Thereforebeginalign1-frac14+frac17-frac110+cdots&=lim_xto1f(x)\&=int_0^1f'(x),mathrm dx\&=int_0^1frac11-x^3,mathrm dx.endalign






    share|cite|improve this answer









    $endgroup$

















      2












      $begingroup$

      If $lvert xrvert<1$, let$$f(x)=sum_n=0^inftyfracx^3n+13n+1.$$Then $$f'(x)=sum_n=0^infty x^3n=frac11-x^3.$$Thereforebeginalign1-frac14+frac17-frac110+cdots&=lim_xto1f(x)\&=int_0^1f'(x),mathrm dx\&=int_0^1frac11-x^3,mathrm dx.endalign






      share|cite|improve this answer









      $endgroup$















        2












        2








        2





        $begingroup$

        If $lvert xrvert<1$, let$$f(x)=sum_n=0^inftyfracx^3n+13n+1.$$Then $$f'(x)=sum_n=0^infty x^3n=frac11-x^3.$$Thereforebeginalign1-frac14+frac17-frac110+cdots&=lim_xto1f(x)\&=int_0^1f'(x),mathrm dx\&=int_0^1frac11-x^3,mathrm dx.endalign






        share|cite|improve this answer









        $endgroup$



        If $lvert xrvert<1$, let$$f(x)=sum_n=0^inftyfracx^3n+13n+1.$$Then $$f'(x)=sum_n=0^infty x^3n=frac11-x^3.$$Thereforebeginalign1-frac14+frac17-frac110+cdots&=lim_xto1f(x)\&=int_0^1f'(x),mathrm dx\&=int_0^1frac11-x^3,mathrm dx.endalign







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Apr 13 at 16:15









        José Carlos SantosJosé Carlos Santos

        175k24134243




        175k24134243



























            draft saved

            draft discarded
















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …


            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.

            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3186397%2fhow-does-this-infinite-series-1-frac14-frac17-frac110-cdots-si%23new-answer', 'question_page');

            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            getting Checkpoint VPN SSL Network Extender working in the command lineHow to connect to CheckPoint VPN on Ubuntu 18.04LTS?Will the Linux ( red-hat ) Open VPNC Client connect to checkpoint or nortel VPN gateways?VPN client for linux machine + support checkpoint gatewayVPN SSL Network Extender in FirefoxLinux Checkpoint SNX tool configuration issuesCheck Point - Connect under Linux - snx + OTPSNX VPN Ububuntu 18.XXUsing Checkpoint VPN SSL Network Extender CLI with certificateVPN with network manager (nm-applet) is not workingWill the Linux ( red-hat ) Open VPNC Client connect to checkpoint or nortel VPN gateways?VPN client for linux machine + support checkpoint gatewayImport VPN config files to NetworkManager from command lineTrouble connecting to VPN using network-manager, while command line worksStart a VPN connection with PPTP protocol on command linestarting a docker service daemon breaks the vpn networkCan't connect to vpn with Network-managerVPN SSL Network Extender in FirefoxUsing Checkpoint VPN SSL Network Extender CLI with certificate

            Image
            If human space travel is limited by the G force vulnerability, is there a way to counter G forces? Did Shadowfax go to Valinor? Do I have a twin with permutated remainders? Blender 2.8 I can't see vertices, edges or faces in edit mode 1960's book about a plague that kills all white people How can I make my BBEG immortal short of making them a Lich or Vampire? Alternative to sending password over mail? Is it inappropriate for a student to attend their mentor's dissertation defense? Why is Collection not simply treated as Collection<?> Took a trip to a parallel universe, need help deciphering Assassin's bullet with mercury Why is it a bad idea to hire a hitman to eliminate most corrupt politicians? I Accidentally Deleted a Stock Terminal Theme Is there a hemisphere-neutral way of specifying a season? What about the virus in 12 Monkeys? Why can't we play rap on piano? Arrow those variables! Today is the Center Modeling an IP Ad...
            Read more

            Cannot Extend partition with GParted The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern) 2019 Community Moderator Election ResultsCan't increase partition size with GParted?GParted doesn't recognize the unallocated space after my current partitionWhat is the best way to add unallocated space located before to Ubuntu 12.04 partition with GParted live?I can't figure out how to extend my Arch home partition into free spaceGparted Linux Mint 18.1 issueTrying to extend but swap partition is showing as Unknown in Gparted, shows proper from fdiskRearrange partitions in gparted to extend a partitionUnable to extend partition even though unallocated space is next to it using GPartedAllocate free space to root partitiongparted: how to merge unallocated space with a partition

            Image
            Make it rain characters Keeping a retro style to sci-fi spaceships? How to split my screen on my Macbook Air? How can I define good in a religion that claims no moral authority? does high air pressure throw off wheel balance? Scientific Reports - Significant Figures Relations between two reciprocal partial derivatives? How to delete random line from file using Unix command? How did passengers keep warm on sail ships? How are presidential pardons supposed to be used? Are my PIs rude or am I just being too sensitive? He got a vote 80% that of Emmanuel Macron’s Change bounding box of math glyphs in LuaTeX Match Roman Numerals Was credit for the black hole image misattributed? How to pronounce 1ターン? how can a perfect fourth interval be considered either consonant or dissonant? Working through the single responsibility principle (SRP) in Python when calls are expensive High Q peak in frequency response means what in time domain? ...
            Read more

            Marilyn Monroe Ny fiainany manokana | Jereo koa | Meny fitetezanafanitarana azy.

            Image
            OlonaBiôgrafiaVangovangoMpilalaoMpilalao amin'ny horonantsaryMpamoaka horonantsaryTeraka tamin'ny taona 1926Maty tamin'ny taona 1962 EtazoniaLos Angeles fanitarana azy . Marilyn Monroe Avy amin'i Wikipedia Sauter à la navigation Sauter à la recherche Marilyn Monroe Ankapobeny Anarana Marilyn Monroe Teraka 1 Jiona 1926 Maty 5 Aogositra 1962 Fiaviana sy ny andraikitra Firenena Etazonia Asa : mpilalao mpilalao amin'ny horonantsary mpamoaka horonantsary Fiainana manokana Marilyn Monroe dia mpilalao, mpilalao amin'ny horonantsary, mpamoaka horonantsary mizaka ny zom-pirenen'i Etazonia teraka ny 1 Jiona 1926 tao Los Angeles ary maty ny 5 Aogositra 1962 Ny fiainany manokana | Ny vadiny dia James Dougherty, Joe DiMaggio, Arthur Miller. Jereo koa | Biôgrafia Rohy ivelany | Ao amin'i Freebase: [1] Mbola ambangovangony ity lahatsoratra ity ary tokony hofenoina . Azonao atao ny mandray anjara eto amin'ny Wikipedia amin...
            Read more