# Does $\lim\limits_{x\to 1^{-}}\sum_{n=1}^{\infty}\frac{x^n}{n(n+1)}$ exist?

by Mike   Last Updated May 08, 2018 05:20 AM

Please, I have a challenge in solving this question. Can anyone help me?

I want to know if $\lim\limits_{x\to 1^{-}}\sum_{n=1}^{\infty}\frac{x^n}{n(n+1)}$ exists and also find it.

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Hint ::

Write the limit as $$\lim_{x\to 1^-} \left(\sum_{r=1}^{\infty} \frac {x^n}{n}\right) -\frac 1x\left(\sum_{r=1}^{\infty} \frac {x^{n+1}}{n+1}\right)$$

Now use that $$-\ln(1-x)=\sum_{r=1}^{\infty} \frac {x^n}{n}$$ and $$-x-\ln(1-x)=\sum_{r=1}^{\infty} \frac {x^{n+1}}{n+1}$$

Manthanein
May 08, 2018 05:01 AM