Chapter Review

Printed Page 630

Definitions:

1. A sequence is a function whose domain is the set of positive integers and whose range is a subset of real numbers. (p. 538)
2. $n$th term of a sequence (p. 538)
3. Limit of a sequence (p. 541)
4. Convergence; divergence of a sequence (p. 541)
5. Related function of a sequence (p. 543)
6. Divergence of a sequence to infinity (p. 545)
7. Bounded sequence (p. 546)
8. Monotonic sequence (p. 548)

Properties of a Convergent Sequence: (p. 542)

If $\{ s_{n}\}$ and $\{ t_{n}\}$ are convergent sequences and if $c$ is a number, then

1. Constant multiple property: $\lim\limits_{n\,\rightarrow \,\infty }\left( cs_{n}\right) =c\lim\limits_{n\,\rightarrow \,\infty }s_{n}$
2. Sum and difference properties: $\lim\limits_{n\,\rightarrow \,\infty }\left( s_{n}\pm t_{n}\right) =\lim\limits_{n\,\rightarrow \,\infty }s_{n}$ $\pm \lim\limits_{~n\,\rightarrow \,\infty }t_{n}$
3. Product property: $\lim\limits_{~n\,\rightarrow \,\infty }\left( s_{n}\cdot t_{n}\right) =\left( \lim\limits_{n\,\rightarrow \,\infty }s_{n}\right) \left( \lim\limits_{n\,\rightarrow \,\infty }t_{n}\right)$
4. Quotient property: $\lim\limits_{~n\,\rightarrow \,\infty }\dfrac{ s_{n}}{t_{n}}=\dfrac{\lim\limits_{n\,\rightarrow \,\infty }s_{n}}{ \lim\limits_{n\,\rightarrow \,\infty }t_{n}}$ provided $\lim\limits_{n\,\rightarrow \,\infty }t_{n}\neq 0$
5. Power property: $\lim\limits_{n\rightarrow \infty }s_{n}^{p}=\left[ \lim\limits_{n\rightarrow \infty }s_{n}\right] ^{p}$ where $p\geq 2$ is an integer
6. Root property: $\lim\limits_{n\rightarrow \infty }\sqrt[p]{s_{n}}=\sqrt [p]{\lim\limits_{n\rightarrow \infty }s_{n}}$, where $p\geq 2$ and $s_{n}\geq 0$ if $p$ is even

Theorems:

1. Let $\{ s_{n}\}$ be a sequence of real numbers. If $\lim\limits_{n\rightarrow \infty }s_{n}=L$ and if $f$ is a function that is continuous at $L$ and is defined for all numbers $s_{n},$ then $\lim\limits_{n\rightarrow \infty }f\!\!\left( s_{n}\right) =f(L) .$ (p. 543)
2. The Squeeze Theorem for sequences (p. 545)
3. A convergent sequence is bounded (p. 547)
4. If a sequence is not bounded from above or if it is not bounded from below, then it diverges. (p. 549)
5. An increasing (or nondecreasing) sequence $\{ s_{n}\}$ that is bounded from above converges. (p. 549)
6. A decreasing (or nonincreasing) sequence $\{ s_{n}\}$ that is bounded from below converges. (p. 549)
7. Let $\{ s_{n}\}$ be a sequence and let $f$ be a related function of $\{ s_{n}\}$. Suppose $L$ is a real number.
   If $\lim\limits_{x\,\rightarrow \,\infty }~f(x)=L$, then $\lim\limits_{n\,\rightarrow \,\infty }s_{n}$ $=L$. (p. 544)
8. The sequence $\{ r^{n}\}$, where $r$ is a real number,

1. converges to $0,$ for $-1<r<1$.
2. converges to $1,$ for $r=1$.
3. diverges for all other numbers (p. 546).

Procedure: Ways to show a sequence is monotonic (p. 548)

Summary: How to determine if a sequence converges (p. 550)

1. If $a_{1}$, $a_{2}$, $\ldots ,$ $a_{n},$ $\ldots$ is an infinite collection of numbers, the expression $\sum\limits_{k\,=\,1}^{\infty}a_{k}=a_{1}+a_{2}+\cdots +a_{n}+\cdots$ is called an infinite series or, simply, a series. (p. 554)
2. $n$th term or general term of a series (p. 554)
3. Partial sum $S_{n}=\sum\limits_{k\,=\,1}^{n}a_{k},$ where $S_{n}$ is the sum of the first $n$ terms of the series $\sum\limits_{k\,=\,1}^{\infty}a_{k}$ (p. 554)
4. Convergence, divergence of a series (p. 555)
5. Geometric series $\sum\limits_{k\,=\,1}^{\infty }ar^{k-1}=a+ar+ar^{2}+\cdots, a\neq 0$ (p. 557)
   $\sum\limits_{k\,=\,1}^{\infty }ar^{k-1}$ converges if $\vert r\vert <1$, and its sum is $\dfrac{a}{1-r}$ $\sum\limits_{k\,=\,1}^{\infty }ar^{k-1}$ diverges if $\vert r\vert \geq 1$. (p. 558)
6. Harmonic series $\sum\limits_{k\,=\,1}^{\infty }\dfrac{1}{k}=1+\dfrac{ 1}{2}+\dfrac{1}{3}+\cdots$ (p. 561)
   The harmonic series diverges. (p. 561)

Summary: Series and convergence of series (p. 561)

1. If the series $\sum\limits_{k\,=\,1}^{\infty }a_{k}$ converges, then $\lim\limits_{n\,\rightarrow \,\infty }a_{n}=0$. (p. 566)
2. The Test for Divergence: The infinite series $\sum\limits_{k\,=\,1}^{ \infty }a_{k}$ diverges if $\lim\limits_{n\,\rightarrow \,\infty}\,a_{n}\neq 0.$ (p. 566)
3. If two infinite series are identical after a certain term, then either both series converge or both series diverge. If both series converge, they do not necessarily have the same sum. (p. 567)
4. The General Convergence Test (p. 569)
5. The Integral Test (p. 569)
6. A $p$-series $\sum\limits_{k=1}^{ \infty }\dfrac{1}{k^{p}} =1+\dfrac{1}{2^{p}} + \dfrac{1}{3^{p}}+\cdots +\dfrac{1 }{n^{p}}+\cdots ,$ where $p$ is a positive real number. (p. 570) The $p$ -series $\sum\limits_{k=1}^{\infty }\dfrac{1}{k^{p}}$ converges if $p>1$ and diverges if $0<p\leq 1$. (p. 571)
7. Bounds on the sum of a $p$-series: If $p>1$, then $\dfrac{1}{p-1} <\sum\limits_{k\,=\,1}^{\infty }$ $\dfrac{1}{k^{p}}<1+\dfrac{1}{p-1}$. (p. 572)

Properties of Convergent Series: If $\sum\limits_{k\,=\,1}^{\infty }a_{k}$ and $\sum\limits_{k\,=\,1}^{\infty }b_{k}$ are two convergent series and if $c\neq 0$ is a number, then

1. Sum and difference properties: \[ \sum\limits_{k\,=\,1}^{\infty }(a_{k}\pm b_{k})=\sum\limits_{k\,=\,1}^{\infty }a_{k}\pm \sum\limits_{k\,=\,1}^{\infty }b_{k} (p. 568) \]

2. 631

   Constant multiple property: \[ \sum\limits_{k\,=\,1}^{\infty}(ca_{k})=c\sum\limits_{k\,=\,1}^{\infty }a_{k} (p. 568) \]
3. If $\sum\limits_{k\,=\,1}^{\infty }a_{k}$ diverges, then $\sum\limits_{k\,=\,1}^{\infty}(ca_{k})$ also diverges. (p. 568)

Theorems:

1. Comparison Test for Convergence: If $0<a_{k}\leq b_{k}$, for all $k,$ and $\sum\limits_{k=1}^{\infty }b_{k}$ converges, then $\sum\limits_{k=1}^{\infty}a_{k}$ converges. (p. 576)
2. Comparison Test for Divergence: If $0<c_{k}\leq a_{k}$, for all $k,$ and $\sum\limits_{k=1}^{\infty }c_{k}$ diverges, then $\sum\limits_{k=1}^{ \infty }a_{k}$ diverges. (p. 576)
3. Limit Comparison Test: Suppose $\sum\limits_{k=1}^{\infty }\,a_{k}$ and $\sum\limits_{k=1}^{\infty }\,b_{k}$ are both series of positive terms. If $\lim\limits_{n\,\rightarrow \,\infty }\,\dfrac{a_{n}}{b_{n}}=L$, $0<L<\infty ,$ then both series converge or both diverge. (p. 577)

Summary: Table 3: Series often used for comparisons (p. 579)

Definitions:

1. Alternating series (p. 582)
2. A series $\sum\limits_{k=1}^{\infty }a_{k}$ is absolutely convergent if the series $\sum\limits_{k\,=\,1}^{\infty }\vert a_{k}\vert$ is convergent. (p. 585)
3. A series that is convergent without being absolutely convergent is conditionally convergent. (p. 587)

Theorems:

1. Alternating Series Test: (p. 582)
2. Error estimate (p. 584)
3. Absolute Convergence Test: If a series $\sum\limits_{k=1}^{\infty }a_{k}$ is absolutely convergent, then it is convergent. (p. 586)

Properties of Absolutely Convergent and Conditionally Convergent series: (p. 588)

1. The alternating harmonic series $\sum\limits_{k=1}^{\infty }\dfrac{(-1)^{k+1}}{k}$ converges. (p. 583).

1. Ratio Test (p. 591)
2. Root Test (p. 593)

1. Guide for choosing a test (p. 597)
2. Tests for convergence and divergence (Table 5; pp. 597-598)

Definitions:

1. Power series: $\sum\limits_{k=0}^{\infty }a_{k}x^{k}$ or $\sum\limits_{k=0}^{\infty }a_{k}(x-c)^{k},$ where $c$ is a constant. (p. 600)
2. Radius of convergence (p. 602)
3. Interval of convergence (p. 602)

Theorems:

1. If a power series centered at 0 converges for a number $x_{0}\neq 0$, then it converges absolutely for all numbers $x$ for which $\vert x\vert <\vert x_{0}\vert$. (p. 602)
2. If a power series centered at 0 diverges for a number $x_{1}$, then it diverges for all numbers $x$ for which $\vert x\vert>\vert x_{1}\vert$. (p. 602)
3. For a power series centered at $c$, exactly one of the following is true (p. 602):

1. The series converges for only $x=c$.
2. The series converges absolutely for all $x$.
3. There is a positive number $R$ for which the series converges absolutely for all $x,$ $\vert x-c \vert <R,$ and diverges for all $x,$ $\vert x-c \vert >R$.

Properties of Power Series: (p. 606)

Let $f(x) =\sum\limits_{k\,=0}^{\infty }a_{k}\,x^{k}$ be a power series in $x$ having a nonzero radius of convergence $R$.

1. Continuity property: $\lim\limits_{x\,\rightarrow \,x_{0}}\left( \sum\limits_{k\,=\,0}^{\infty }a_{k}\,x^{k}\right) =\sum\limits_{k\,=\,0}^{\infty }\left( \lim\limits_{x\,\rightarrow \,x_{0}}a_{k}\,x^{k}\right)$ $=\sum\limits_{k\,=\,0}^{\infty }a_{k}\,x_{0}^{k}$
2. Differentiation property: $\dfrac{d}{dx}\left( \sum\limits_{k\,=\,0}^{\infty }a_{k}\,x^{k}\right) =\sum\limits_{k\,=\,0}^{\infty }\left( \dfrac{d}{dx}a_{k}\,x^{k}\right)$ $=\sum\limits_{k\,=\,1}^{\infty }k\,a_{k}\,x^{k-1}$
3. Integration property: $\int_{0}^{x}\left( \sum\limits_{k\,=\,0}^{\infty }a_{k}\,t^{k}\right) dt=\sum\limits_{k\,=\,0}^{\infty }\left( \int_{0}^{x}a_{k}\,t^{k}~dt\right)$ $=\sum\limits_{k\,=\,0}^{\infty }\dfrac{a_{k}\,x^{k+1}}{k+1}$

Theorems:

1. Taylor series: (p. 613) $$\begin{eqnarray*} f(x) &=& f\,(c)+f^{\prime} (c)(x-c)+\dfrac{f^{\prime \prime} (c) }{2!}(x-c) ^{2}\\ &&+\cdots +\dfrac{f\,^{(n) }(c)\,}{n!}(x-c)^{n}{+}\cdots =\sum\limits_{k\,=\,0}^{\infty }\dfrac{f^{(k) }(c) }{k!}(x-c) ^{k}\quad \end{eqnarray*}$$
2. Maclaurin series (p. 613) $f(x)=f(0)+f^{\prime} (0)\,x+\dfrac{ f^{\prime \prime} (0) \,x^{2}}{2!}+\cdots +\dfrac{ f^{(n)}(0)\,x^{n}}{n!}+\cdots =\sum\limits_{k\,=\,0}^{\infty }\dfrac{ f^{(k) }(0) }{k!}x^{k}$
3. Taylor’s formula with remainder (p. 614)
4. Convergence of a Taylor series (p. 615)
5. Binomial series (p. 620)
6. Convergence of a binomial series (p. 620)

(pp. 623-628)