If $ \displaystyle\sum_{n=1}^{\infty} a_n $ is convergent then discuss the convergence or divergence of the following series whose $n^{th}$ term is

  • $ a_n \sin n $
  • $\displaystyle \frac {a_n}{1+| a_n |}$

If $\displaystyle \sum _{n=1}^{\infty} a_n $ is absolutely convergent then then $\displaystyle \sum_{n=1}^{\infty} a_n \sin n $ and $\displaystyle \sum _{n=1}^{\infty} \dfrac {a_n}{1+|a_n|} $ are convergent. But what if $\displaystyle \sum _{n=1}^{\infty} a_n $ is conditionally convergent. A little hint would be appreciated. Thanks in advance.

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    $\begingroup$ @suchandaadhikari try $a_n = \frac{\sin(n)}{n}$, and you can see here for the proof that $\frac{\sin^2(n)}{n}$ doesn't converge $\endgroup$ – Jakobian Dec 8 '18 at 13:09
  • $\begingroup$ @ jakobian thank you so much ..nice example. .can u give a hint about the 2nd series. . $\endgroup$ – suchanda adhikari Dec 8 '18 at 13:58
  • $\begingroup$ I think other series is convergent as both have same nature as $n\rightarrow \infty$ $\endgroup$ – neelkanth Dec 8 '18 at 13:59
  • $\begingroup$ If you please elaborate a little more... $\endgroup$ – suchanda adhikari Dec 8 '18 at 14:06
  • $\begingroup$ @Jakobian Why not an official answer? $\endgroup$ – Paul Frost Dec 8 '18 at 16:13

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