Question

Sum to infinity the series whose $n^{th}$ term is $\frac{(-1{)}^{n+1}{x}^n}{n(n+1)(n+2)}$.

Answer 1

$\Rightarrow 2S=(\frac{1}{1}-\frac{2}{2}+\frac{1}{3})x-(\frac{1}{2}-\frac{2}{3}+\frac{1}{4})x^2+(\frac{1}{3}-\frac{2}{4}+\frac{1}{5})x^3+....$

$=(x-\frac{x^2}{2}+\frac{x^3}{3})-\frac{2}{x}(\frac{x^2}{2}-\frac{x^3}{3}+\frac{x^4}{4})+\frac{1}{x^2}(\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}-....)$

$=\log (1+x)-\frac{2}{x}(x-\log (1+x))+\frac{1}{x^2}\{-x+\frac{x^2}{2}+\log (1+x)\}$

$=(1+\frac{2}{x}+\frac{1}{x^2})\log (1+x)-\frac{3}{2}-\frac{1}{x}$