Sum of power series $\sum_{n=1}^\infty (-1)^n\frac{n(n+1)}{2^n}x^n$

Calculate the sum of series:

$$\sum_{n=1}^\infty (-1)^n\frac{n(n+1)}{2^n}x^n$$

I know how to calculate sum of power series, but I don’t know what should I do with $(-1)^n$

Answer

Hint:

Try first to find what is
$$
\sum_{n=1}^\infty n (n+1)y^n
$$
by noting that
$$
n(n+1)y^n=(y^{n+1})”\cdot y
$$

A most important technique about calculation of power series is differentiation (and integration) term by term, which should be discussed in any serious real (complex) analysis textbook. See also a note by Gowers.


Besides the “formal” calculation, another issue you still need to address is that for what $y$ the series is convergent.

Attribution
Source : Link , Question Author : Yas , Answer Author : Community

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