Chapter 4: Problem 3
Use the comparison test to determine whether the following series converge. $$ \sum_{n=1}^{\infty} \frac{1}{2(n+1)} $$
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The following advanced exercises use a generalized ratio test to determine convergence of some series that arise in particular applications when tests in this chapter, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if \(\lim _{n \rightarrow \infty} \frac{a_{2 n}}{a_{n}}<1 / 2\), then \(\sum a_{n}\) converges, while if \(\lim _{n \rightarrow \infty} \frac{a_{2 n+1}}{a_{n}}>1 / 2\), then \(\sum a_{n}\) diverges. Let \(a_{n}=\frac{\pi^{\ln n}}{(\ln n)^{n}} .\) Show that \(\frac{a_{2 n}}{a_{n}} \rightarrow 0\) as \(n \rightarrow \infty\).
For the following exercises, indicate whether each of the following statements is true or false. If the statement is false, provide an example in which it is false.Let \(a_{n}^{+}=a_{n}\) if \(a_{n} \geq 0\) and \(a_{n}^{-}=-a_{n}\) if \(a_{n}<0 .\) (Also, \(a_{n}^{+}=0\) if \(a_{n}<0\) and \(a_{n}^{-}=0\) if \(\left.a_{n} \geq 0 .\right)\) If \(\sum_{n=1}^{\infty} a_{n}\) converges conditionally but not absolutely, then neither \(\sum_{n=1}^{\infty} a_{n}^{+}\) nor \(\sum_{n=1}^{\infty} a_{n}^{-}\) converge.
Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit. $$ a_{n}=\frac{2^{n+1}}{5^{n}} $$
Use the ratio test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, or state if the ratio test is inconclusive. $$ \sum_{n=1}^{\infty} \frac{2^{n^{2}}}{n^{n} n !} $$
Does there exist a number \(p\) such that \(\sum_{n=1}^{\infty} \frac{2^{n}}{n^{p}}\) converges?
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