91Ó°ÊÓ

Does the series $$ \sum_{k=1}^{\infty} \log \left(\frac{k+1}{k}\right) $$ converge or diverge?

Let \(\left\\{a_{n}\right\\}\) be a sequence of real numbers converging to zero. Show that there must exist a monotonic sequence \(\left\\{b_{n}\right\\}\) such that the series \(\sum_{k=1}^{\infty} b_{k}\) diverges and the series \(\sum_{k=1}^{\infty} a_{k} b_{k}\) is absolutely convergent.

Obtain a closed form for $$ \sum_{k=1}^{n} \frac{1}{k(k+2)(k+4)} $$

Let \(\left\\{a_{n}\right\\}\) be a sequence of positive numbers and write $$ L_{n}=\frac{\log \left(\frac{1}{a_{n}}\right)}{\log n} $$ Show that if \(\lim \inf L_{n}>1\), then \(\sum a_{n}\) converges. Show that if \(L_{n} \leq 1\) for all sufficiently large \(n\), then \(\sum a_{n}\) diverges.

Suppose that \(\sum_{n=1}^{\infty} a_{n}^{2}\) converges. Show that $$ \limsup _{n \rightarrow \infty} \frac{a_{1}+\sqrt{2} a_{2}+\sqrt{3} a_{3}+\sqrt{4} a_{4}+\cdots+\sqrt{n} a_{n}}{n}<\infty $$

Obtain a closed form for $$ \sum_{k=1}^{n} \frac{\alpha r+\beta}{k(k+1)(k+2)} $$

Let \(x_{1}, x_{2}, x_{3}\) be a sequence of positive numbers and write $$ s_{n}=\frac{x_{1}+x_{2}+x_{3}+\cdots+x_{n}}{n} $$ and $$ t_{n}=\frac{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\cdots+\frac{1}{x_{n}}}{n} $$ If \(s_{n} \rightarrow S\) and \(t_{n} \rightarrow T\), show that \(S T \geq 1\).

Show that $$ \frac{1}{r-1}=\frac{1}{r+1}+\frac{2}{r^{2}+1}+\frac{4}{r^{4}+1}+\frac{8}{r^{8}+1}+\ldots $$ for all \(r>1\).

Prove the alternating series test directly from the Cauchy criterion.

Let \(\left\\{a_{k}\right\\}\) and \(\left\\{b_{k}\right\\}\) be sequences with \(\left\\{b_{k}\right\\}\) decreasing and $$ \left|a_{1}+a_{2}+\cdots+a_{k}\right| \leq K $$ for all \(k\). Show that $$ \left|\sum_{k=1}^{n} a_{k} b_{k}\right| \leq K b_{1} $$ for all \(n\).