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.