<s>[INST] Compute the sum of the power serie $\sum_{1}^{\infty } \frac{(n^{2} + n)x^{n-1}}{2^{n-1}}$
I have found convergence interval for series: $\left | x \right | < 2$,
but I have no idea how to find the sum of the power series.
 [/INST] $$\sum\limits_{n=1}^\infty{(n^2+n)x^{n-1}\over2^{n-1}} = 4{d^2\over dx^2}\sum\limits_{n=1}^\infty{x^{n+1}\over2^{n+1}} = 4{d^2\over dx^2}{{x^2\over2^2}\over1-{x\over2}} = 2{d^2\over dx^2}{x^2\over2-x} = 2{d^2\over dx^2}{x^2-4+4\over2-x} = 2{d^2\over dx^2}\left(-x-2 + {4\over2-x}\right) = {8\cdot1\cdot2\over(2-x)^3} = {16\over(2-x)^3}$$
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