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Question Number 104176 by Dwaipayan Shikari last updated on 19/Jul/20

\underset{n = 1}{\sum^{\infty}} ({(\frac{n}{n + 1})}^{2} - \frac{2}{n + 1} - 1)

Answered by OlafThorendsen last updated on 19/Jul/20

u_{n} = {(\frac{n}{n + 1})}^{2} - \frac{2}{n + 1} - 1

u_{n} = - \frac{4 n + 3}{{(n + 1)}^{2}}

u_{n} = - \frac{4}{n + 1} + \frac{1}{{(n + 1)}^{2}}

\underset{n = 1 (}{\sum^{\infty}} \frac{1}{{n + 1)}^{2}} is convergent

\underset{n = 1}{\sum^{\infty}} \frac{4}{n + 1} is divergent

\Rightarrow \underset{n = 1}{\sum^{\infty}} u_{n} is divergent

(sorry for my approximative english)