급수 심화 전체 문제

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.

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

The formula $\cos \theta =1-\frac{{\theta }^{2}}{2!}+\frac{{\theta }^{4}}{4!}-\frac{{\theta }^{6}}{6!}+\cdots$ will be derived in the next chapter. Use the remainder $|{R}_{N}|\le {b}_{N+1}$ to find a bound for the error in estimating $\cos \theta$ by the fifth partial sum $1-{\theta }^{2}/2!+{\theta }^{4}/4!-{\theta }^{6}/6!+{\theta }^{8}/8!$ for $\theta =1,$ $\theta =\pi /6,$ and $\theta =\pi .$

The formula $\sin \theta =\theta -\frac{{\theta }^{3}}{3!}+\frac{{\theta }^{5}}{5!}-\frac{{\theta }^{7}}{7!}+\cdots$ will be derived in the next chapter. Use the remainder $|{R}_{N}|\le {b}_{N+1}$ to find a bound for the error in estimating $\sin \theta$ by the fifth partial sum $\theta -{\theta }^{3}/3!+{\theta }^{5}/5!-{\theta }^{7}/7!+{\theta }^{9}/9!$ for $\theta =1,$ $\theta =\pi /6,$ and $\theta =\pi .$

How many terms in $\cos \theta =1-\frac{{\theta }^{2}}{2!}+\frac{{\theta }^{4}}{4!}-\frac{{\theta }^{6}}{6!}+\cdots$ are needed to approximate $\cos 1$ accurate to an error of at most $0.00001?$

How many terms in $\sin \theta =\theta -\frac{{\theta }^{3}}{3!}+\frac{{\theta }^{5}}{5!}-\frac{{\theta }^{7}}{7!}+\cdots$ are needed to approximate $\sin 1$ accurate to an error of at most $0.00001?$

Sometimes the alternating series $\sum_{n=1}^{\infty }{(-1)}^{n-1}{b}_{n}$ converges to a certain fraction of an absolutely convergent series $\sum_{n=1}^{\infty }{b}_{n}$ at a faster rate. Given that $\sum_{n=1}^{\infty }\frac{1}{{n}^{2}}=\frac{{\pi }^{2}}{6},$ find $12=1-\frac{1}{{2}^{2}}+\frac{1}{{3}^{2}}-\frac{1}{{4}^{2}}+\cdots .$ Which of the series $6\sum_{n=1}^{\infty }\frac{1}{{n}^{2}}$ and $S\sum_{n=1}^{\infty }\frac{{(-1)}^{n-1}}{{n}^{2}}$ gives a better estimation of ${\pi }^{2}$ using $1000$ terms?

The following alternating series converge to given multiples of $\pi .$ Find the value of $N$ predicted by the remainder estimate such that the $N\text{th}$ partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum $N$ for which the error bound holds, and give the desired approximate value in each case. Up to $15$ decimals places, $\pi =3.141592653589793\ldots .$

[T] $\frac{\pi }{4}=\sum_{n=0}^{\infty }\frac{{(-1)}^{n}}{2n+1},$ error $<0.0001$

The following alternating series converge to given multiples of $\pi .$ Find the value of $N$ predicted by the remainder estimate such that the $N\text{th}$ partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum $N$ for which the error bound holds, and give the desired approximate value in each case. Up to $15$ decimals places, $\pi =3.141592653589793\ldots .$

[T] $\frac{\pi }{\sqrt{12}}=\sum_{k=0}^{\infty }\frac{{(-3)}^{-k}}{2k+1},$ error $<0.0001$

The following alternating series converge to given multiples of $\pi .$ Find the value of $N$ predicted by the remainder estimate such that the $N\text{th}$ partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum $N$ for which the error bound holds, and give the desired approximate value in each case. Up to $15$ decimals places, $\pi =3.141592653589793\ldots .$

[T] If $\sum_{n=1}^{N}{(-1)}^{n-1}\frac{1}{n}\to \ln 2,$ what is $1+\frac{1}{3}+\frac{1}{5}-\frac{1}{2}-\frac{1}{4}-\frac{1}{6}+\frac{1}{7}+\frac{1}{9}+\frac{1}{11}-\frac{1}{8}-\frac{1}{10}-\frac{1}{12}+\cdots ?$

[T] Plot the series $\sum_{n=1}^{100}\frac{\cos (2\pi nx)}{n}$ for $0\le x<1.$ Explain why $\sum_{n=1}^{100}\frac{\cos (2\pi nx)}{n}$ diverges when $x=0,1.$ How does the series behave for other $x?$

[T] Plot the series $\sum_{n=1}^{100}\frac{\sin (2\pi nx)}{n}$ for $0\le x<1$ and comment on its behavior

[T] The alternating harmonic series converges because of cancellation among its terms. Its sum is known because the cancellation can be described explicitly. A random harmonic series is one of the form $\sum_{n=1}^{\infty }\frac{{S}_{n}}{n},$ where ${s}_{n}$ is a randomly generated sequence of $\pm 1\text{'s}$ in which the values $\pm 1$ are equally likely to occur. Use a random number generator to produce $1000$ random $\pm 1\text{s}$ and plot the partial sums ${S}_{N}=\sum_{n=1}^{N}\frac{{s}_{n}}{n}$ of your random harmonic sequence for $N=1$ to $1000.$ Compare to a plot of the first $1000$ partial sums of the harmonic series.

[T] The Euler transform rewrites $S=\sum_{n=0}^{\infty }{(-1)}^{n}{b}_{n}$ as $S=\sum_{n=0}^{\infty }{(-1)}^{n}{2}^{-n-1}\sum_{m=0}^{n}\left(\begin{array}{c}n \\ m\end{array}\right){b}_{n-m}.$ For the alternating harmonic series, it takes the form $\ln (2)=\sum_{n=1}^{\infty }\frac{{(-1)}^{n-1}}{n}=\sum_{n=1}^{\infty }\frac{1}{n{2}^{n}}.$ Compute partial sums of $\sum_{n=1}^{\infty }\frac{1}{n{2}^{n}}$ until they approximate $\ln (2)$ accurate to within $0.0001.$ How many terms are needed? Compare this answer to the number of terms of the alternating harmonic series are needed to estimate $\ln (2).$

[T] In the text it was stated that a conditionally convergent series can be rearranged to converge to any number. Here is a slightly simpler, but similar, fact. If ${a}_{n}\ge 0$ is such that ${a}_{n}\to 0$ as $n\to \infty$ but $\sum_{n=1}^{\infty }{a}_{n}$ diverges, then, given any number $A$ there is a sequence ${s}_{n}$ of $\pm 1\text{'s}$ such that $\sum_{n=1}^{\infty }{a}_{n}{s}_{n}\to A.$ Show this for $A>0$ as follows.

a. Recursively define ${s}_{n}$ by ${s}_{n}=1$ if ${S}_{n-1}=\sum_{k=1}^{n-1}{a}_{k}{s}_{k}<A$ and ${s}_{n}=-1$ otherwise. b. Explain why eventually ${S}_{n}\ge A,$ and for any $m$ larger than this $n,$ $A-{a}_{m}\le {S}_{m}\le A+{a}_{m}.$ c. Explain why this implies that ${S}_{n}\to A$ as $n\to \infty .$

Use the ratio test to determine whether $\sum_{n=1}^{\infty }{a}_{n}$ converges, where ${a}_{n}$ is given in the following problems. State if the ratio test is inconclusive.

${a}_{n}=1/n!$