91Ó°ÊÓ

For which \(r>0\) does the series \(\sum_{n=1}^{\infty} \frac{r^{n}}{2^{n}}\) converge?

For which \(r>0\) does the series \(\sum_{n=1}^{\infty} \frac{2^{n}}{r^{n}}\) converge?

Does \(\sum_{n=1}^{\infty} \frac{\sin ^{2}(n r / 2)}{n}\) converge or diverge? Explain.

Suppose that \(a_{n} \geq 0\) and \(b_{n} \geq 0\) and that \(\sum_{n=1}^{\infty} a_{n}^{2}\) and \(\sum_{n=1}^{\infty} b_{n}^{2}\) converge. Prove that \(\sum_{n=1}^{\infty} a_{n} b_{n}\) converges and \(\sum_{n=1}^{\infty} a_{n} b_{n} \leq \frac{1}{2}\left(\sum_{n=1}^{\infty} a_{n}^{2}+\sum_{n=1}^{\infty} b_{n}^{2}\right)\).

Does \(\sum_{n=1}^{\infty} 2^{-\ln \ln n}\) converge? (Hint: Write \(2^{\ln \ln n}\) as a power of \(\ln n .\) )

Does \(\sum_{n=1}^{\infty}(\ln n)^{-\ln n}\) converge? (Hint: Use \(t=e^{\ln (t)}\) to compare to a \(p\) - series.)

Does \(\sum_{n=2}^{\infty}(\ln n)^{-\ln \ln n}\) converge? (Hint: Compare \(a_{n}\) to \(\left.1 / n .\right)\)

Show that if \(a_{n} \geq 0\) and \(\sum_{n=1}^{\infty} a_{n}\) converges, then \(\sum_{n=1}^{\infty} a_{n}^{2}\) converges. If \(\sum_{n=1}^{\infty} a_{n}^{2}\) converges, does \(\sum_{n=1}^{\infty} a_{n}\) necessarily converge?

Suppose that \(a_{n}>0\) for all \(n\) and that \(\sum_{n=1}^{\infty} a_{n}\) converges. Suppose that \(b_{n}\) is an arbitrary sequence of zeros and ones. Does \(\sum_{n=1}^{\infty} a_{n} b_{n}\) necessarily converge?

Suppose that \(a_{n}>0\) for all \(n\) and that \(\sum_{n=1}^{\infty} a_{n}\) diverges. Suppose that \(b_{n}\) is an arbitrary sequence of zeros and ones with infinitely many terms equal to one. Does \(\sum_{n=1}^{\infty} a_{n} b_{n}\) necessarily diverge?