급수의 수렴과 발산

Express the following series as a telescoping sum and evaluate its nth partial sum.

$\sum_{n=1}^{\infty }\frac{(n+2)}{n(n+1){2}^{n+1}}$ (Hint: Look at $1/(n{2}^{n}).)$

A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms.

${a}_{n}=f(n)-f(n+1)-f(n+2)+f(n+3),$ in which $f(n)\to 0$ as $n\to \infty .$ Find $\sum_{n=1}^{\infty }{a}_{n}.$

Suppose that ${a}_{n}={c}_{0}f(n)+{c}_{1}f(n+1)+{c}_{2}f(n+2)+{c}_{3}f(n+3)+{c}_{4}f(n+4),$ where $f(n)\to 0$ as $n\to \infty .$ Find a condition on the coefficients ${c}_{0},\ldots ,{c}_{4}$ that make this a general telescoping series.

A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms.

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

A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms.

Evaluate $\sum_{n=2}^{\infty }\frac{2}{{n}^{3}-n}.$

A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms.

Find a formula for $\sum_{n=1}^{\infty }\frac{1}{n(n+N)}$ where $N$ is a positive integer.

Each of the following infinite series converges to the given multiple of $\pi$ or $1/\pi .$

In each case, find the minimum value of $N$ such that the $N\text{th}$ partial sum of the series accurately approximates the left-hand side to the given number of decimal places, and give the desired approximate value. Up to $15$ decimals place, $\pi =3.141592653589793....$

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

Each of the following infinite series converges to the given multiple of $\pi$ or $1/\pi .$

In each case, find the minimum value of $N$ such that the $N\text{th}$ partial sum of the series accurately approximates the left-hand side to the given number of decimal places, and give the desired approximate value. Up to $15$ decimals place, $\pi =3.141592653589793....$

[T] $\frac{\pi }{2}=\sum_{k=0}^{\infty }\frac{k!}{(2k+1)!!}=\sum_{k=0}^{\infty }\frac{{2}^{k}k{!}^{2}}{(2k+1)!},$ error $<{10}^{-4}$

Each of the following infinite series converges to the given multiple of $\pi$ or $1/\pi .$

In each case, find the minimum value of $N$ such that the $N\text{th}$ partial sum of the series accurately approximates the left-hand side to the given number of decimal places, and give the desired approximate value. Up to $15$ decimals place, $\pi =3.141592653589793....$

[T] $\frac{9801}{2\pi }=\frac{\sqrt{2}}{9801}\sum_{k=0}^{\infty }\frac{(4k)!(1103+26390k)}{{(k!)}^{4}{396}^{4k}},$ error $<{10}^{-12}$

Each of the following infinite series converges to the given multiple of $\pi$ or $1/\pi .$

In each case, find the minimum value of $N$ such that the $N\text{th}$ partial sum of the series accurately approximates the left-hand side to the given number of decimal places, and give the desired approximate value. Up to $15$ decimals place, $\pi =3.141592653589793....$

[T] $\frac{1}{12\pi }=\sum_{k=0}^{\infty }\frac{{(-1)}^{k}(6k)!(13591409+545140134k)}{(3k)!{(k!)}^{3}{640320}^{3k+3/2}},$ error $<{10}^{-15}$

[T] A fair coin is one that has probability $1/2$ of coming up heads when flipped.

a. What is the probability that a fair coin will come up tails $n$ times in a row? b. Find the probability that a coin comes up heads for the first time on the last of an even number of coin flips.

[T] Find the probability that a fair coin is flipped a multiple of three times before coming up heads.

[T] The expected number of times that a fair coin will come up heads is defined as the sum over $n=1,2,\ldots$ of $n$ times the probability that the coin will come up heads exactly $n$ times in a row, or $n/{2}^{n+1}.$ Compute the expected number of consecutive times that a fair coin will come up heads.

Suppose that ${a}_{n}\ge 0$ is a sequence of numbers. Explain why the sequence of partial sums of ${a}_{n}$ is increasing.

[T] Suppose that ${a}_{n}$ is a sequence of positive numbers and the sequence ${S}_{n}$ of partial sums of ${a}_{n}$ is bounded above. Explain why $\sum_{n=1}^{\infty }{a}_{n}$ converges. Does the conclusion remain true if we remove the hypothesis ${a}_{n}\ge 0?$

[T] Suppose that ${a}_{1}={S}_{1}=1$ and that, for given numbers $S>1$ and $0<k<1,$ one defines ${a}_{n+1}=k(S-{S}_{n})$ and ${S}_{n+1}={a}_{n+1}+{S}_{n}.$ Does ${S}_{n}$ converge? If so, to what? (Hint: First argue that ${S}_{n}<S$ for all $n$ and ${S}_{n}$ is increasing.)

[T] A version of von Bertalanffy growth can be used to estimate the age of an individual in a homogeneous species from its length if the annual increase in year $n+1$ satisfies ${a}_{n+1}=k(S-{S}_{n}),$ with ${S}_{n}$ as the length at year $n,$ $S$ as a limiting length, and $k$ as a relative growth constant. If ${S}_{1}=3,$ $S=9,$ and $k=1/2,$ numerically estimate the smallest value of $n$ such that ${S}_{n}\ge 8.$ Note that ${S}_{n+1}={S}_{n}+{a}_{n+1}.$ Find the corresponding $n$ when $k=1/4.$

[T] Suppose that $\sum_{n=1}^{\infty }{a}_{n}$ is a convergent series of positive terms. Explain why $\lim_{N\to \infty }\sum_{n=N+1}^{\infty }{a}_{n}=0.$

In Exercises 21–30, state whether the given series converges or diverges.

$\displaystyle \sum_{n=1}^\infty \frac{1}{n^5}$

In Exercises 21–30, state whether the given series converges or diverges.

$\displaystyle \sum_{n=0}^\infty \frac{1}{5^n}$