91Ó°ÊÓ

State whether each statement is true, or give an example to show that it is false. If \(\sum_{n=1}^{\infty} a_{n} x^{n}\) converges, then \(a_{n} x^{n} \rightarrow 0\) as \(n \rightarrow \infty\).

State whether each statement is true, or give an example to show that it is false. \(\sum_{n=1}^{\infty} a_{n} x^{n}\) converges at \(x=0\) for any real numbers \(a_{n}\).

State whether each statement is true, or give an example to show that it is false. Given any sequence \(a_{n}\), there is always some \(R>0,\) possibly very small, such that \(\sum_{n=1}^{\infty} a_{n} x^{n}\) converges on \((-R, R)\).

State whether each statement is true, or give an example to show that it is false. If \(\sum_{n=1}^{\infty} a_{n} x^{n}\) has radius of convergence \(R>0\) and if \(\left|b_{n}\right| \leq\left|a_{n}\right|\) for all \(n\), then the radius of convergence of \(\sum_{n=1}^{\infty} b_{n} x^{n}\) is greater than or equal to \(R\).

State whether each statement is true, or give an example to show that it is false. Suppose that \(\sum_{n=0}^{\infty} a_{n}(x+1)^{n}\) converges at \(x=-2\). At which of the following points must the series also converge? Use the fact that if \(\sum a_{n}(x-c)^{n}\) converges at \(x\), then it converges at any point closer to \(c\) than \(x\). a. \(\quad x=2\) b. \(\quad x=-1\) c. \(\quad x=-3\) d. \(\quad x=0\) e. \(x=0.99\) f. \(\quad x=0.000001\)

Suppose that \(\left|\frac{a_{n+1}}{a_{n}}\right| \rightarrow 1\) as \(n \rightarrow \infty\). Find the radius of convergence for each series. $$ \sum_{n=0}^{\infty} a_{n} 2^{n} x^{n} $$

Suppose that \(\left|\frac{a_{n+1}}{a_{n}}\right| \rightarrow 1\) as \(n \rightarrow \infty\). Find the radius of convergence for each series. $$ \sum_{n=0}^{\infty} \frac{a_{n} x^{n}}{2^{n}} $$

Suppose that \(\left|\frac{a_{n+1}}{a_{n}}\right| \rightarrow 1\) as \(n \rightarrow \infty\). Find the radius of convergence for each series. $$ \sum_{n=0}^{\infty} \frac{a_{n} \pi^{n} x^{n}}{e^{n}} $$

Suppose that \(\left|\frac{a_{n+1}}{a_{n}}\right| \rightarrow 1\) as \(n \rightarrow \infty\). Find the radius of convergence for each series. $$ \sum_{n=0}^{\infty} \frac{a_{n}(-1)^{n} x^{n}}{10^{n}} $$

Suppose that \(\left|\frac{a_{n+1}}{a_{n}}\right| \rightarrow 1\) as \(n \rightarrow \infty\). Find the radius of convergence for each series. $$ \sum_{n=0}^{\infty} a_{n}(-4)^{n} x^{2 n} $$