교대급수

[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 $n\to \infty .$

Why is $\displaystyle \sum_{n=1}^\infty \sin n$ not an alternating series?

A series $\displaystyle \sum_{n=1}^\infty (-1)^na_n$ converges when $\{a_n\}$ is , _____ and $\displaystyle \lim_{n\to\infty}a_n =$.

Give an example of a series where $\displaystyle \sum_{n=0}^\infty a_n$ converges but $\displaystyle \sum_{n=0}^\infty |a_n|$ does not.

The sum of a _____ convergent series can be changed by rearranging the order of its terms.

In Exercises 5–20, an alternating series $\displaystyle \sum_{n=i}^\infty a_n$ is given.

(a) Determine if the series converges or diverges. (b) Determine if $\displaystyle \sum_{n=0}^\infty |a_n|$ converges or diverges. (c) If $\displaystyle \sum_{n=0}^\infty a_n$ converges, determine if the convergence is conditional or absolute.

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

In Exercises 5–20, an alternating series $\displaystyle \sum_{n=i}^\infty a_n$ is given.

(a) Determine if the series converges or diverges. (b) Determine if $\displaystyle \sum_{n=0}^\infty |a_n|$ converges or diverges. (c) If $\displaystyle \sum_{n=0}^\infty a_n$ converges, determine if the convergence is conditional or absolute.

$\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n+1}}{\sqrt{n!}}$

In Exercises 5–20, an alternating series $\displaystyle \sum_{n=i}^\infty a_n$ is given.

(a) Determine if the series converges or diverges. (b) Determine if $\displaystyle \sum_{n=0}^\infty |a_n|$ converges or diverges. (c) If $\displaystyle \sum_{n=0}^\infty a_n$ converges, determine if the convergence is conditional or absolute.

$\displaystyle \sum_{n=0}^\infty (-1)^{n}\frac{n+5}{3n-5}$

In Exercises 5–20, an alternating series $\displaystyle \sum_{n=i}^\infty a_n$ is given.

(a) Determine if the series converges or diverges. (b) Determine if $\displaystyle \sum_{n=0}^\infty |a_n|$ converges or diverges. (c) If $\displaystyle \sum_{n=0}^\infty a_n$ converges, determine if the convergence is conditional or absolute.

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

In Exercises 5–20, an alternating series $\displaystyle \sum_{n=i}^\infty a_n$ is given.

(a) Determine if the series converges or diverges. (b) Determine if $\displaystyle \sum_{n=0}^\infty |a_n|$ converges or diverges. (c) If $\displaystyle \sum_{n=0}^\infty a_n$ converges, determine if the convergence is conditional or absolute.

$\displaystyle \sum_{n=0}^\infty (-1)^{n+1}\frac{3n+5}{n^2-3n+1}$

In Exercises 5–20, an alternating series $\displaystyle \sum_{n=i}^\infty a_n$ is given.

(a) Determine if the series converges or diverges. (b) Determine if $\displaystyle \sum_{n=0}^\infty |a_n|$ converges or diverges. (c) If $\displaystyle \sum_{n=0}^\infty a_n$ converges, determine if the convergence is conditional or absolute.

$\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n}}{\ln n+1}$

In Exercises 5–20, an alternating series $\displaystyle \sum_{n=i}^\infty a_n$ is given.

(a) Determine if the series converges or diverges. (b) Determine if $\displaystyle \sum_{n=0}^\infty |a_n|$ converges or diverges. (c) If $\displaystyle \sum_{n=0}^\infty a_n$ converges, determine if the convergence is conditional or absolute.

$\displaystyle \sum_{n=2}^\infty (-1)^n\frac{n}{\ln n }$

In Exercises 5–20, an alternating series $\displaystyle \sum_{n=i}^\infty a_n$ is given.

(a) Determine if the series converges or diverges. (b) Determine if $\displaystyle \sum_{n=0}^\infty |a_n|$ converges or diverges. (c) If $\displaystyle \sum_{n=0}^\infty a_n$ converges, determine if the convergence is conditional or absolute.

$\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n+1}}{1+3+5+\cdots+(2n-1) }$

In Exercises 5–20, an alternating series $\displaystyle \sum_{n=i}^\infty a_n$ is given.

(a) Determine if the series converges or diverges. (b) Determine if $\displaystyle \sum_{n=0}^\infty |a_n|$ converges or diverges. (c) If $\displaystyle \sum_{n=0}^\infty a_n$ converges, determine if the convergence is conditional or absolute.

$\displaystyle \sum_{n=1}^\infty \cos \big(\pi n\big)$