Chapter 5

Since \(\sum_{2}^{\infty} 1 /(n \log n)\) diverges, \(\lim _{n \rightarrow \infty} \sum_{2}^{n} 1 /(k \log k)=\infty .\) How many terms must be taken before the partial sums exceed \(10 ?\)

Investigate the convergence of the following series: (a) \(\frac{1}{3}+\frac{2}{6}+\frac{3}{11}+\frac{4}{18}+\frac{5}{27}+\cdots\) (b) \(\frac{1}{2}-\frac{2}{20}+\frac{3}{38}-\frac{4}{56}+\frac{5}{74}-\cdots\) (c) \(\frac{1}{3}+\frac{1 \cdot 2}{3 \cdot 5}+\frac{1 \cdot 2 \cdot 3}{3 \cdot 5 \cdot 7}+\cdots\) (d) \(\frac{1}{4}+\frac{1 \cdot 9}{4 \cdot 16}+\frac{1 \cdot 9 \cdot 25}{4 \cdot 16 \cdot 36}+\frac{1 \cdot 9 \cdot 25 \cdot 49}{4 \cdot 16 \cdot 36 \cdot 64}+\cdots\)

Form the Cauchy product of the following series: $$ \begin{aligned} &1+2+4+8+16+32+\cdots \\ &1-1+1-1+1-1+1-1+\cdots \end{aligned} $$ and find a formula for the coefficients of the resulting series.

Show that if \(\sum a_{n}\) converges, then \(\sum_{N}^{\infty} a_{n} \rightarrow 0\) as \(N \rightarrow \infty\)

Find a formula for the coefficients of the Cauchy product of the series \(\sum_{0}^{\infty} A^{n}\) and \(\sum_{0}^{\infty} B^{n} .\)

Investigate the convergence of \(\sum a_{n}\) where (a) \(a_{n}=\frac{\sqrt{n+1}-\sqrt{n}}{n+1}\) (b) \(a_{n}=\sqrt{\frac{\sqrt{n+1}-\sqrt{n}}{n+1}}\)

If \(\sum_{0}^{\infty} a_{n} x^{n}=\left(\sum_{0}^{\infty} x^{n}\right)\left(\sum_{0}^{\infty} x^{2 n}\right)\), what is \(a_{n}\) ?

Show that the sum of an alternating series lies between any paar of successive partial sums, so that the error made in stopping at the nth term does not exceed the absolute value of the next term.

Estimate the sum of each of the following series, accurate to \(.005\) : (a) \(\sum_{1}^{x}(-1)^{n} n / 10^{n}\) (b) \(\sum_{1}^{x}(-1)^{n+1} \frac{1}{n^{3}}\)

(a)\( Let \)00\(, then \)\sum a_{n}$ diverges.

For which values of \(r\) and \(s\) does the double series \(\sum \sum_{m, n=1}^{\infty} r^{m} s^{n}\) converge?

Let \(\sum a_{n}\) and \(\sum b_{n}\) converge, with \(b_{n}>0\) for all \(n\). Suppose that \(a_{n} / b_{n} \rightarrow L\). Prove that $$ \sum_{N}^{\infty} a_{k} / \sum_{N}^{\infty} b_{k} \rightarrow L $$

Let \(f(x) \geq 0, f^{\prime}(x) \geq 0, f^{\prime \prime}(x) \geq 0\) for \(1 \leq x<\infty\). Show that $$ 0 \leq \sum_{1}^{n} f(k)-\int_{1}^{n} f-\frac{1}{2} f(n)-\frac{1}{2} f(1) \leq \frac{1}{4} f^{\prime}(n) \text { for } n \geq 1 . $$

Formal algebra yields the expression: $$ (1+x)^{\rho}=1+p x+\frac{p(p-1)}{2} x^{2}+\cdots+\left(\begin{array}{l} p \\ n \end{array}\right) x^{n}+\cdots $$

Let \(\left\\{a_{n}\right\\} \downarrow 0 ;\) show that \(\sum_{1}^{\infty} a_{n}\) converges if and only if \(\sum_{1}^{\infty} 2^{n} a_{2^{n}}\) converges.

Investigate the convergence of the series \(\sum \sum_{k, n=1}^{\infty} 1 /(n+3)^{2 k}\)

Some of the following statements are true and some are false; prove those that are true, and disprove those that are false. \((a)\) If \(\sum a_{n}\) and \(\sum b_{n}\) converge, so does \(\sum\left(a_{n}+b_{n}\right)\). (b) If \(\sum a_{n}\) and \(\sum b_{n}\) diverge, so does \(\sum\left(a_{n}+b_{n}\right)\). (c) If \(\sum\left|a_{n}\right|\) is convergent, so is \(\sum\left(a_{n}\right)^{2}\). \((d)\) If \(\sum\left|a_{n}\right|\) and \(\sum\left|b_{n}\right|\) converge so does \(\sum a_{n} b_{n}\). (e) If \(\sum_{1}^{\infty} a_{n}^{2}\) converges, so does \(\sum_{1}^{\infty} a_{n} / n\). \({ }^{*}(f)\) If \(\left\\{a_{n}\right\\} \downarrow 0\) and \(\sum a_{n}\) converges, then \(\lim _{n \rightarrow \infty} n a_{n}=0\)

Show that \(\sum_{2}^{x} 1 /\left(n^{2}+3 n-4\right)=137 / 300\).

Show that if \(f \geq 0\) and \(f\) is monotonically decreasing, and if \(c_{n}=\sum_{1}^{n} f(k)-\int_{1}^{n} f(x) d x\), then \(\lim _{n \rightarrow \infty} c_{n}\) exists.

Show that if \(c_{n} \geq 0\) and \(\sum c_{n}\) converges, then \(\sum \sqrt{c_{n}} / n\) also converges.

Discuss the convergence of \(\sum_{n=1}^{\infty} \frac{(3 n) !}{(n !)^{3}} x^{n}\).

Show that if \(a_{n}>0, \sum a_{n}\) diverges, and \(S_{n}=a_{1}+\cdots+a_{n}\), then \(\sum a_{n} / S_{n}\) also diverges, but more slowly.

\({\) Let \(f\) and \(f^{\prime}\) be continuous on the interval \(1 \leq x<\infty\) with \(f(x)>0\) and $$ \int_{1}^{\infty}\left|f^{\prime}(x)\right| d x $$ convergent. Show that the series \(\sum_{1}^{\infty} f(k)\) and the improper integral \(\int_{1}^{\infty} f(x) d x\) are either both convergent or both divergent.

Show that if \(\sum a_{n}^{2} / n\) converges, then \(1 / N \sum_{1}^{N} a_{k} \rightarrow 0\).

Show that a reasonable value for this unending product is \(1 / 2 .\) $$ \left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{16}\right)\left(1-\frac{1}{25}\right)\left(1-\frac{1}{36}\right) \cdots $$

What is a reasonable value to assign to the unending expression: $$ \sqrt{2+\sqrt{2}+\sqrt{2+\sqrt{2}+\sqrt{2}+}} $$