mathematics

Problems in summation of series

Series: Infinite Series

Problem 1 Find a closed form solution for $f(z) = \sum_{n=1}^\infty \sum_{k=1}^n \frac{1}{k}z^n = z + \frac{3}{2}z^2 + \frac{11}{6}z^3 + \dots$. Solution We have $$ \begin{align*} f(z) - zf(z) &= z + \frac{z^2}{2} + \frac{z^3}{3} + \dots \
&= \int \frac{1}{1-z}dz = -ln(1-z) \
\implies f(z) &= -\frac{ln(1-z)}{1-z} \end{align*} $$ Problem 2 Evaluate $\sum_{n=0}^{\infty} \frac{(-1)^n)}{3^n(2n + 1)}$.