Basic Analysis I — Real Analysis · Problem 2.6.5

Sequences and Series — Jiri Lebl

Mathematics — Real Analysis · power-series-radius-of-convergence

Problem

For the following power series, find if they are convergent or not, and if so find their radius of convergence.

a)\; \sum_{n=0}^{\infty} 2^n x^n \qquad b)\; \sum_{n=0}^{\infty} n x^n \qquad c)\; \sum_{n=0}^{\infty} n!\, x^n \qquad d)\; \sum_{n=0}^{\infty} \frac{1}{(2n)!}(x-10)^n \qquad e)\; \sum_{n=0}^{\infty} x^{2n} \qquad f)\; \sum_{n=0}^{\infty} n!\, x^{n!}