급수 심화 전체 문제

Show that for the remainder estimate to apply on $[N,\infty )$ it is sufficient that $f(x)$ be decreasing on $[N,\infty ),$ but $f$ need not be decreasing on $[1,\infty ).$

Suppose that ${a}_{n}\ge 0$ and ${b}_{n}\ge 0$ and that $\sum_{n=1}^{\infty }{a}^{2}{}_{n}$ and $\sum_{n=1}^{\infty }{b}^{2}{}_{n}$ converge. Prove that $\sum_{n=1}^{\infty }{a}_{n}{b}_{n}$ converges and $\sum_{n=1}^{\infty }{a}_{n}\,{b}_{n}\le \frac{1}{2}\left(\sum_{n=1}^{\infty }{a}_{n}^{2}+\sum_{n=1}^{\infty }{b}_{n}^{2}\right).$

Show that if ${a}_{n}\ge 0$ and $\sum_{n=1}^{\infty }{a}_{n}$ converges, then $\sum_{n=1}^{\infty }{a}^{2}{}_{n}$ converges. If $\sum_{n=1}^{\infty }{a}^{2}{}_{n}$ converges, does $\sum_{n=1}^{\infty }{a}_{n}$ necessarily converge?

Use the limit comparison test to determine whether each of the following series converges or diverges.

Show that if ${a}_{n}\ge 0$ and $\sum_{n=1}^{\infty }{a}^{2}{}_{n}$ converges, then $\sum_{n=1}^{\infty }{\sin }^{2}({a}_{n})$ converges.

Suppose that ${a}_{n}/{b}_{n}\to 0$ in the comparison test, where ${a}_{n}\ge 0$ and ${b}_{n}\ge 0.$ Prove that if $\sum {b}_{n}$ converges, then $\sum {a}_{n}$ converges.

In view of the previous exercise, it may be surprising that a subseries of the harmonic series in which about one in every five terms is deleted might converge. A depleted harmonic series is a series obtained from $\sum_{n=1}^{\infty }\frac{1}{n}$ by removing any term $1/n$ if a given digit, say $9,$ appears in the decimal expansion of $n.$ Argue that this depleted harmonic series converges by answering the following questions.

a. How many whole numbers $n$ have $d$ digits? b. How many $d\text{-digit}$ whole numbers $h(d).$ do not contain $9$ as one or more of their digits? c. What is the smallest $d\text{-digit}$ number $m(d)?$ d. Explain why the deleted harmonic series is bounded by $\sum_{d=1}^{\infty }\frac{h(d)}{m(d)}.$ e. Show that $\sum_{d=1}^{\infty }\frac{h(d)}{m(d)}$ converges.

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?

[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.)

Let ${a}_{n}={2}^{-[n/2]}$ where $[x]$ is the greatest integer less than or equal to $x.$ Determine whether $\sum_{n=1}^{\infty }{a}_{n}$ converges and justify your answer.

Let ${a}_{n}=\frac{1}{1+x}\,\frac{2}{2+x}\cdots \frac{n}{n+x}\,\frac{1}{n}=\frac{(n-1)!}{(1+x)(2+x)\cdots (n+x)}.$ Show that ${a}_{2n}/{a}_{n}\le {e}^{-x/2}/2.$ For which $x>0$ does the generalized ratio test imply convergence of $\sum_{n=1}^{\infty }{a}_{n}?$ (Hint: Write $2{a}_{2n}/{a}_{n}$ as a product of $n$ factors each smaller than $1/(1+x/(2n)).)$

The following advanced exercises use a generalized ratio test to determine convergence of some series that arise in particular applications when tests in this chapter, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if $\lim_{n\to \infty }\frac{{a}_{2n}}{{a}_{n}}<1/2,$ then $\sum {a}_{n}$ converges, while if $\lim_{n\to \infty }\frac{{a}_{2n+1}}{{a}_{n}}>1/2,$ then $\sum {a}_{n}$ diverges.

Let ${a}_{n}=\frac{{n}^{\ln \,n}}{{(\ln \,n)}^{n}}.$ Show that $\frac{{a}_{2n}}{{a}_{n}}\to 0$ as $n\to \infty .$

In the following exercises, state whether each statement is true, or give an example to show that it is false.

If $\sum_{n=1}^{\infty }{a}_{n}{x}^{n}$ converges, then ${a}_{n}{x}^{n}\to 0$ as $n\to \infty .$

In the following exercises, state whether each statement is true, or give an example to show that it is false.

$\sum_{n=1}^{\infty }{a}_{n}{x}^{n}$ converges at $x=0$ for any real numbers ${a}_{n}.$

[T] Evaluate the power series expansion $\ln (1+x)=\sum_{n=1}^{\infty }{(-1)}^{n-1}\frac{{x}^{n}}{n}$ at $x=1$ to show that $\ln (2)$ is the sum of the alternating harmonic series. Use the alternating series test to determine how many terms of the sum are needed to estimate $\ln (2)$ accurate to within 0.001, and find such an approximation.

Show that, up to powers x³ and y³, $E(x)=\sum_{n=0}^{\infty }\frac{{x}^{n}}{n!}$ satisfies $E(x+y)=E(x)E(y).$

Differentiate the series $E(x)=\sum_{n=0}^{\infty }\frac{{x}^{n}}{n!}$ term-by-term to show that $E(x)$ is equal to its derivative.

Show that if $f(x)=\sum_{n=0}^{\infty }{a}_{n}{x}^{n}$ is a sum of even powers, that is, ${a}_{n}=0$ if n is odd, then $F={\int }_{0}^{x}f(t)dt$ is a sum of odd powers, while if f is a sum of odd powers, then F is a sum of even powers.

In the following exercises, verify that the given choice of n in the remainder estimate $|{R}_{n}|\le \frac{M}{(n+1)!}{(x-a)}^{n+1},$ where M is the maximum value of $|{f}^{(n+1)}(z)|$ on the interval between a and the indicated point, yields $|{R}_{n}|\le \frac{1}{1000}.$ Find the value of the Taylor polynomial p_n of $f$ at the indicated point. If $|Rn|$ is not less than $1/1000$, determine what it is.

[T] $\sqrt{10};a=9,n=3$