Recitation 22

Example 1: Find the radius of convergence and interval of convergence of the series. (a**)** $\sum_{n=1}^\infty\frac{x^n}{n!}$; (b) $\sum_{n=1}^\infty n!(2x-1)^n$; (c) $\sum_{n=1}^\infty\frac{n^2x^n}{2\cdot4\cdot6\cdot\ldots\cdot(2n)}$.

Hint: Use the ratio test.

Example 2: If $\sum_{n=0}^\infty c_n4^n$ is convergent, does it follow that the following series are convergent? (a) $\sum_{n=0}^\infty c_n(-2)^n$; (b) $\sum_{n=0^\infty}c_n(-4)^n$.

Solution: (a) Because the radius of convergence of $\sum_{n=0}^\infty c_nx^n$ is at least $4$, it (absolutely) converges for $x=-2$. (b) If the radius of convergence is exactly $4$, then its convergence for $x=-4$ is inconclusive.