급수 심화 전체 문제

Show that the alternating series $1-\frac{1}{2}+\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{6}+\frac{1}{4}-\frac{1}{8}+\cdots$ does

not converge. What hypothesis of the alternating series test is not met?

Suppose that $\sum {a}_{n}$ converges absolutely. Show that the series consisting of the positive terms ${a}_{n}$ also converges.

Show that the alternating series $\frac{2}{3}-\frac{3}{5}+\frac{4}{7}-\frac{5}{9}+\cdots$ does not converge. What hypothesis of the alternating series test is not met?

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 $<0.0001$

[T] The series $\sum_{n=0}^{\infty }\frac{\sin (x+\pi n)}{x+\pi n}$ plays an important role in signal processing. Show that $\sum_{n=0}^{\infty }\frac{\sin (x+\pi n)}{x+\pi n}$ converges whenever $0<x<\pi .$ (Hint: Use the formula for the sine of a sum of angles.)

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] Plot the series $\sum_{n=1}^{100}\frac{\cos (2\pi nx)}{{n}^{2}}$ for $0\le x<1$ and describe its graph.

[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] Estimates of $\sum_{n=1}^{\infty }\frac{1}{{n}^{2}}$ can be accelerated by writing its partial sums as $\sum_{n=1}^{N}\frac{1}{{n}^{2}}=\sum_{n=1}^{N}\frac{1}{n(n+1)}+\sum_{n=1}^{N}\frac{1}{{n}^{2}(n+1)}$ and recalling that $\sum_{n=1}^{N}\frac{1}{n(n+1)}=1-\frac{1}{N+1}$ converges to one as $N\to \infty .$ Compare the estimate of ${\pi }^{2}/6$ using the sums $\sum_{n=1}^{1000}\frac{1}{{n}^{2}}$ with the estimate using $1+\sum_{n=1}^{1000}\frac{1}{{n}^{2}(n+1)}.$

[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.

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