**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.