급수 심화 전체 문제

State whether each of the following series converges absolutely, conditionally, or not at all.

$\sum_{n=1}^{\infty }\frac{1}{n}\sin \left(\frac{n\pi }{2}\right)$

State whether each of the following series converges absolutely, conditionally, or not at all.

$\sum_{n=1}^{\infty }\sin (n\pi /2)\sin (1/n)$

In each of the following problems, use the estimate $|{R}_{N}|\le {b}_{N+1}$ to find a value of $N$ that guarantees that the sum of the first $N$ terms of the alternating series $\sum_{n=1}^{\infty }{(-1)}^{n+1}{b}_{n}$ differs from the infinite sum by at most the given error. Calculate the partial sum ${S}_{N}$ for this $N.$

[T] ${b}_{n}=1/n,$ error $<{10}^{-5}$

In each of the following problems, use the estimate $|{R}_{N}|\le {b}_{N+1}$ to find a value of $N$ that guarantees that the sum of the first $N$ terms of the alternating series $\sum_{n=1}^{\infty }{(-1)}^{n+1}{b}_{n}$ differs from the infinite sum by at most the given error. Calculate the partial sum ${S}_{N}$ for this $N.$

[T] ${b}_{n}=1/\ln (n),$ $n\ge 2,$ error $<{10}^{-1}$

In each of the following problems, use the estimate $|{R}_{N}|\le {b}_{N+1}$ to find a value of $N$ that guarantees that the sum of the first $N$ terms of the alternating series $\sum_{n=1}^{\infty }{(-1)}^{n+1}{b}_{n}$ differs from the infinite sum by at most the given error. Calculate the partial sum ${S}_{N}$ for this $N.$

[T] ${b}_{n}=1/\sqrt{n},$ error $<{10}^{-3}$

In each of the following problems, use the estimate $|{R}_{N}|\le {b}_{N+1}$ to find a value of $N$ that guarantees that the sum of the first $N$ terms of the alternating series $\sum_{n=1}^{\infty }{(-1)}^{n+1}{b}_{n}$ differs from the infinite sum by at most the given error. Calculate the partial sum ${S}_{N}$ for this $N.$

[T] ${b}_{n}=1/{2}^{n},$ error $<{10}^{-6}$

In each of the following problems, use the estimate $|{R}_{N}|\le {b}_{N+1}$ to find a value of $N$ that guarantees that the sum of the first $N$ terms of the alternating series $\sum_{n=1}^{\infty }{(-1)}^{n+1}{b}_{n}$ differs from the infinite sum by at most the given error. Calculate the partial sum ${S}_{N}$ for this $N.$

[T] ${b}_{n}=\ln \left(1+\frac{1}{n}\right),$ error $<{10}^{-3}$

In each of the following problems, use the estimate $|{R}_{N}|\le {b}_{N+1}$ to find a value of $N$ that guarantees that the sum of the first $N$ terms of the alternating series $\sum_{n=1}^{\infty }{(-1)}^{n+1}{b}_{n}$ differs from the infinite sum by at most the given error. Calculate the partial sum ${S}_{N}$ for this $N.$

[T] ${b}_{n}=1/{n}^{2},$ error $<{10}^{-6}$

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.

If ${b}_{n}\ge 0$ is decreasing and $\lim_{n\to \infty }{b}_{n}=0,$ then $\sum_{n=1}^{\infty }({b}_{2n-1}-{b}_{2n})$ converges absolutely.

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.

If ${b}_{n}\ge 0$ is decreasing, then $\sum_{n=1}^{\infty }({b}_{2n-1}-{b}_{2n})$ converges absolutely.

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.

If ${b}_{n}\ge 0$ and $\lim_{n\to \infty }{b}_{n}=0$ then $\sum_{n=1}^{\infty }(\frac{1}{2}({b}_{3n-2}+{b}_{3n-1})-{b}_{3n})$ converges.

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.

If ${b}_{n}\ge 0$ is decreasing and $\sum_{n=1}^{\infty }({b}_{3n-2}+{b}_{3n-1}-{b}_{3n})$ converges then $\sum_{n=1}^{\infty }{b}_{3n-2}$ converges.

If ${b}_{n}\ge 0$ is decreasing and $\sum_{n=1}^{\infty }{(-1)}^{n-1}{b}_{n}$ converges conditionally but not absolutely, then ${b}_{n}$ does not tend to zero.

Let ${a}_{n}^{+}={a}_{n}$ if ${a}_{n}\ge 0$ and ${a}_{n}^{-}=-{a}_{n}$ if ${a}_{n}<0.$ (Also, ${a}_{n}^{+}=0\,\text{if}\,{a}_{n}<0$ and ${a}_{n}^{-}=0\,\text{if}\,{a}_{n}\ge 0.)$ 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.

Suppose that ${a}_{n}$ is a sequence of positive real numbers and that $\sum_{n=1}^{\infty }{a}_{n}$ converges.

Suppose that ${b}_{n}$ is an arbitrary sequence of ones and minus ones. Does $\sum_{n=1}^{\infty }{a}_{n}{b}_{n}$ necessarily converge?

Suppose that ${a}_{n}$ is a sequence such that $\sum_{n=1}^{\infty }{a}_{n}{b}_{n}$ converges for every possible sequence ${b}_{n}$ of zeros and ones. Does $\sum_{n=1}^{\infty }{a}_{n}$ converge absolutely?

The following series do not satisfy the hypotheses of the alternating series test as stated.

In each case, state which hypothesis is not satisfied. State whether the series converges absolutely.

$\sum_{n=1}^{\infty }{(-1)}^{n+1}\frac{{\sin }^{2}n}{n}$

The following series do not satisfy the hypotheses of the alternating series test as stated.

In each case, state which hypothesis is not satisfied. State whether the series converges absolutely.

$\sum_{n=1}^{\infty }{(-1)}^{n+1}\frac{{\cos }^{2}n}{n}$

The following series do not satisfy the hypotheses of the alternating series test as stated.

In each case, state which hypothesis is not satisfied. State whether the series converges absolutely.

$1+\frac{1}{2}-\frac{1}{3}-\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\frac{1}{7}-\frac{1}{8}+\cdots$

The following series do not satisfy the hypotheses of the alternating series test as stated.

In each case, state which hypothesis is not satisfied. State whether the series converges absolutely.

$1+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}+\frac{1}{8}-\frac{1}{9}+\cdots$