급수의 수렴과 발산

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

$\sum_{n=1}^{\infty }\frac{2n+1}{{({n}^{2}+n)}^{2}}$ (Hint: Factor denominator and use partial fractions.)

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

$\sum_{n=2}^{\infty }\frac{\ln \left(1{+}_{n}^{1}\right)}{\ln n\ln (n+1)}$

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.

[T] Define a sequence ${t}_{k}=\sum_{n=1}^{k-1}(1/n)-\ln k.$ Use the graph of $1/x$ to verify that ${t}_{k}$ is increasing. Plot ${t}_{k}$ for $k=1\ldots 100$ and state whether it appears that the sequence converges.

[T] Suppose that $N$ equal uniform rectangular blocks are stacked one on top of the other, allowing for some overhang. Archimedes’ law of the lever implies that the stack of $N$ blocks is stable as long as the center of mass of the top $(N-1)$ blocks lies at the edge of the bottom block. Let $x$ denote the position of the edge of the bottom block, and think of its position as relative to the center of the next-to-bottom block. This implies that $(N-1)x=\left(\frac{1}{2}-x\right)$ or $x=1/(2N).$ Use this expression to compute the maximum overhang (the position of the edge of the top block over the edge of the bottom block.) See the following figure.

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] Find a series that expresses the probability that a fair coin will come up heads for the second time on a multiple of three flips.

[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.

[T] A certain drug is effective for an average patient only if there is at least $1$ mg per kg in the patient’s system, while it is safe only if there is at most $2$ mg per kg in an average patient’s system. Suppose that the amount in a patient’s system diminishes by a multiplicative factor of $0.9$ each hour after a dose is administered. Find the maximum interval $N$ of hours between doses, and corresponding dose range $d$ (in mg/kg) for this $N$ that will enable use of the drug to be both safe and effective in the long term.

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