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Showing that $sum_n=1^inftyfraca_na_n+b_n$ converges. [duplicate]

The Next CEO of Stack Overflowhow prove $sum_n=1^inftyfraca_nb_n+a_n $is convergent?prove series converges$frac a_n+1a_n le frac b_n+1b_n$ If $sum_n=1^infty b_n$ converges then $sum_n=1^infty a_n$ converges as wellIf $sum a_n$ converges and $b_n=sumlimits_k=n^inftya_n $, prove that $sum fraca_nb_n$ divergesIf $sum a_n b_n$ converges for all $(b_n)$ such that $b_n to 0$, then $sum |a_n|$ converges.$sumlimits_n=1^infty a_n^2$ and $sumlimits_n=1^infty b_n^2$ converge show $sumlimits_n=1^infty a_n b_n$ converges absolutelyProve if $sumlimits_n=1^ infty a_n$ converges, $b_n$ is bounded & monotone, then $sumlimits_n=1^ infty a_nb_n$ converges.If $sum_n=0^infty|a_n|^p,sum_n=0^infty|b_n|^p $ converge then $sum_n=0^infty|a_n+b_n|^p$ convergesA question about real series $sum_n=1^infty a_n$ and $sum_n=1^infty b_n$Show that $sum_n=0^infty(sum_j=0^n a_jb_n-j)$ converges to $(sum_n=0^inftyb_n)(sum_n=0^inftya_n)$.$sum_n=1^infty a_n^b_n$ convergesProve $(a_n,b_n >0) land sum a_n $ converges $ land sum b_n $ diverges$implies liminflimits_nrightarrow infty fraca_nb_n=0$