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

Question

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.

Thanks in advance.

Answer 1

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

Answer 2

The given series converges uniformly on $[0,1]$ by Weierstrass M. Thus the limit of the sum is the sum of the limits. In other words, the limit in question equals

$$\sum_{n=1}^{\infty} \frac{1}{n(n+1)} = \sum_{n=1}^{\infty} \left (\frac{1}{n} - \frac{1}{n+1}\right) = 1,$$

the last series telescoping nicely.