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

Asked Dec 08 '18 at 12:53

Active Dec 08 '18 at 13:46

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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.

edited Dec 08 '18 at 13:04

Yadati Kiran

2,388

asked Dec 08 '18 at 12:53

suchanda adhikari

7

@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 08 '18 at 13:09
@ jakobian thank you so much ..nice example. .can u give a hint about the 2nd series. . – suchanda adhikari Dec 08 '18 at 13:58
I think other series is convergent as both have same nature as $n\rightarrow \infty$ – neelkanth Dec 08 '18 at 13:59
If you please elaborate a little more... – suchanda adhikari Dec 08 '18 at 14:06
@Jakobian Why not an official answer? – Paul Frost Dec 08 '18 at 16:13
@PaulFrost it's not a full answer though. – Jakobian Dec 08 '18 at 18:35

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

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