If $\sum_{n=1}^{\infty} a_n$ is convergent, then discuss the convergence or divergence of the following series.

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.

• @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 – Jakobian Dec 8 '18 at 13:09
• @ jakobian thank you so much ..nice example. .can u give a hint about the 2nd series. . – suchanda adhikari Dec 8 '18 at 13:58
• I think other series is convergent as both have same nature as $n\rightarrow \infty$ – neelkanth Dec 8 '18 at 13:59
• If you please elaborate a little more... – suchanda adhikari Dec 8 '18 at 14:06
• @Jakobian Why not an official answer? – Paul Frost Dec 8 '18 at 16:13