<s>[INST] Finding sum $\frac{1}{2} x^2 + \frac{1}{2} \frac{1}{3} x^3 + \frac{1}{4} \frac{1}{3} x^4 + ... $ I need to find following: for $0 < x < 1$
$$\frac{1}{2} x^2 + \frac{1}{2} \frac{1}{3} x^3 + \frac{1}{4} \frac{1}{3} x^4  + ... $$
My attempt: 
I can see that the sum is composed of two infinite sums, one is $\sum_{n=1}^{\infty} \frac{1}{n(n+1)} = \sum_{n=1}^{\infty} \left( \frac{1}{n} -
 \frac{1}{n+1} \right)$ (Telescoping ) and another is, $\sum_{n=2}^{\infty} x^n$ (it's a G.P.). How can I use these for solving the sum in question?
Any hints will be appreciated...
 [/INST] Hint
U can make the series into a geometric progression where $$a=\frac{x^2}{2}$$  and $$r=(x/(n+1))/(1/(n-1))$$
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