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Series

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• Intro
• Divergence and Integral Test
• Comparison Tests
• Alternating Series
• Ratio and Root Tests

Intro

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• Ratio and Root Tests

Definition (Series)

An infinite series is a sume of infinitely many terms and is written in the form $$\sum_{n=1}^{\infty}a_n = a_1 +a_2 +\cdots$$ and the parital sum of an infinite series is $$S_k = \sum_{n=1}^k a_n = a_1+a_2+\cdots +a_k$$

The partial sum form a sequence $\{S_k\}$. If the sequence of partial sums converges to a real number $S$, the infinite series converges.

If we can describe the convergence of a series to $S$, we call $S$ the sum of the series, and we write

$$\sum_{n=1}^{\infty}a_n=S$$

If the sequence of partial sums diverges, we have the divergence of a series.

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Definition (Harmonic Series)

The harmonic series is defined as $$\sum_{n=1}^{\infty}\frac{1}{n}=1+\frac{1}{2}+\frac13+\cdots$$

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Algebraic Properties of Convergent Series

$\newcommand{\SUM}[2]{\sum_{n=1}^{#1}\left(#2\right)}$ Let $\SUM{\infty}{a_n}$ and $\SUM{\infty}{b_n}$ be convegent series. Then the following algebraic properties hold.

1. The series $\SUM{\infty}{a_n\pm b_n}$ converges and $\SUM{\infty}{a_n\pm b_n}=\SUM{\infty}{a_n}\pm \SUM{\infty}{b_n}$
2. For any real number $c$, the series $\SUM{\infty}{ca_n}$ converges and $\SUM{\infty}{ca_n}=c\SUM{\infty}{a_n}$

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Defintion (Geometric Series)

A geometric series is a series of the form

$$\SUM{\infty}{ar^{n-1}}=a+ar+ar^2+ar^3+\cdots$$

• If $|r|<1$, the eries converges, and

$$\SUM{\infty}{ar^{n-1}}=\frac{a}{1-r}$$

were $|r|<1$.

• If $|r|>1$, the series diverges.

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Definition (Telescoping Series)

A telescoping series is a series in which most of the terms cancel in each of the partial sums, leaving only some of the first terms and some of the last terms. For example: $$\SUM{k}{b_n-b_{n+1}} = b_1-b_{k+1}$$

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sum(1/2,i,1,10);
10*(1/2);
(1/2)^1+(1/2)^2+(1/2)^3+(1/2)^4;
sum((1/2)^n,n,1,4);
plot2d([discrete,makelist((1/2)^i,i,4)],[style,points])$

$$\tag{${\it \%o}_{2}$}5$$

$$\tag{${\it \%o}_{3}$}5$$

$$\tag{${\it \%o}_{4}$}{{15}\over{16}}$$

$$\tag{${\it \%o}_{5}$}{{15}\over{16}}$$

As you can see above instead of evaluating:

$$(1/2)^1+(1/2)^2+(1/2)^3+(1/2)^4 = 15/16$$

we compact the notation to: $$\sum_{n=1}^4 (1/2)^n = 15/16$$

Divergence and Integral Test

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Theorem (Divergence Test)

If $\lim_{n\to \infty}a_n=c\ne 0$ or $\lim_{n\to \infty}a_n$ does not exist, then the series $\SUM{\infty}{a_n}$ diverges.

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Theorem (Integral Test)

Consider the series $\SUM{\infty}{a_n}$.

• If the sequence of $a_n$ is made up of positive terms.
• If there exists a function $f$ and a positive integer $N$ such that the following three conditions are satisfied:

1. $f$ is continuous,
2. $f$ is decreasing, and
3. $f(n)=a_n$ for all integers $n\ge N$.

Then $$\SUM{\infty}{a_n} \text{ and } \int_N^{\infty}f(x)\,dx$$ both converge or both diverge.

NOTE

Very important to satisfy the three conditions in order to use the Integral test.

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Definition ($p$-series)

For any real number $p$, the series $$\SUM{\infty}{\frac{1}{n^p}}$$ is called a $p$-series.

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FACTs ($p$-series Convergence or Divergence)

(More will follow after comparison tests like the ratio test and root test) Given the $p$-series $\SUM{\infty}{\frac{1}{n^p}}$

• If $p<0$, then $1/n^p \to \infty$, and if $p=0$, then $1/n^p \to 1$. Therefore, by the divergence test the $p$-series diverges if $p\le 0$.
• If $p>0$, then $f(x)=1/x^p$ is a positive, continuous, decreasing function (integral test). Therefore, for $p>0$, we use the integral test, comparing $$\SUM{\infty}{\frac{1}{n^p}} \text{ and }\int_1^{\infty}\frac{1}{x^p}\,dx$$ and from improper integrals we know
  • If $p>1$ then the $p$-series converges.
  • if $0<p\le 1$ then the $p$-series diverges.

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Theorem (Remainder Estimate from the Integral Test)

Suppose $\SUM{\infty}{a_n}$ is a convergent series with positive terms. Suppose there exists a function $f$ satisfying the following three conditions:

1. $f$ is continous,
2. $f$ is decreasing, and
3. $f(n)=a_n$ for all integers $n\ge 1$.

Let $S_N$ be the $N^{\text{th}}$ partial sum of $\SUM{\infty}{a_n}$. For all positive integers $N$,

$$S_N+\int_{N+1}^{\infty}f(x)\,dx<\SUM{\infty}{a_n}<S_N+\int_N^{\infty}f(x)\,dx$$

In other words, the remainder $$R_N=\SUM{\infty}{a_n}-S_N = \sum_{n=N+1}^{\infty}\left(a_n\right)$$ satisfies the following estimate:

$$\int_{N+1}^{\infty}f(x)\,dx < R_n < \int_N^{\infty}f(x)\,dx.$$

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Example (Estimating the value of a series)

Consider $\SUM{\infty}{\frac{1}{n^3}}$

1. Find the error for $S_{10}$.
2. Determien the elase value of $N$ necessary such that $S_N$ will estimate $\SUM{\infty}{1/n^3}$ within $0.001$.

sum(1/n^3,n,1,10);
float(sum(1/n^3,n,1,10));
f(x):=1/x^3;
integrate(f(x),x);
assume(N>1)$
integrate(f(x),x,1,N);
limit(integrate(f(x),x,1,N),N,infinity);

$$\tag{${\it \%o}_{7}$}{{19164113947}\over{16003008000}}$$

$$\tag{${\it \%o}_{8}$}1.197531985674193$$

$$\tag{${\it \%o}_{9}$}f\left(x\right):={{1}\over{x^3}}$$

$$\tag{${\it \%o}_{10}$}-{{1}\over{2\,x^2}}$$

$$\tag{${\it \%o}_{12}$}{{1}\over{2}}-{{1}\over{2\,N^2}}$$

$$\tag{${\it \%o}_{13}$}{{1}\over{2}}$$

As we can see from the calculations the remainder estimate is $R_n=\frac{1}{2N^2}$. So if we want the error for $S_{10}$ we have:

Rn(N):=1/(2*N^2);
Rn(10);
float(Rn(10));

$$\tag{${\it \%o}_{14}$}{\it Rn}\left(N\right):={{1}\over{2\,N^2}}$$

$$\tag{${\it \%o}_{15}$}{{1}\over{200}}$$

$$\tag{${\it \%o}_{16}$}0.005$$

Now determine least value of $N$. That is solve the following inequality: $$R_N<0.001 \text{ or } \frac{1}{2N^2}<0.001$$

assume(N>1)$
solve(1/(2*N^2)=0.001,N);
float(solve(1/(2*N^2)=0.001,N));

rat: replaced -0.001 by -1/1000 = -0.001

$$\tag{${\it \%o}_{18}$}\left[ N=-2\,5^{{{3}\over{2}}} , N=2\,5^{{{3}\over{2}}} \right]$$

rat: replaced -0.001 by -1/1000 = -0.001

$$\tag{${\it \%o}_{19}$}\left[ N=-22.3606797749979 , N=22.3606797749979 \right]$$

Therefore, the minimum necessary value is $N=23$.

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Comparison Test

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Theorem (Comparison Test)

1. Suppose there exists an integer $N$ such that $0\le a_n\le b_n$ for all $n\ge N$. If $\SUM{\infty}{b_n}$ converges, then $\SUM{\infty}{a_n}$ converges.
2. Suppose there exists an integer $N$ such that $a_n\ge b_n\ge 0$ for all $n\ge N$. If $\SUM{\infty}{b_n}$ diverges, then $\SUM{\infty}{a_n}$ diverges.

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Theorem (Limit Comparison Test)

Let $a_n$, $b_n\ge 0$ for all $n\ge 1$.

1. If $\lim_{n\to \infty}\left(a_n/b_n\right) = L \ne 0$, then $\SUM{\infty}{a_n}$ and $\SUM{\infty}{b_n}$ both converge or both diverge.
2. If $\lim_{n\to \infty}\left(a_n/b_n\right) = 0$ and $\SUM{\infty}{b_n}$ converges, then $\SUM{\infty}{a_n}$ converges
3. If $\lim_{n\to \infty}\left(a_n/b_n\right) = \infty$ and $\SUM{\infty}{b_n}$ diverges, then $\SUM{\infty}{a_n}$ diverges.

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Alternating Series

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Definition (Alternating Series)

Any series whose terms alternate between positive and negative values is called an alternating series. An alternating series can be written in the form

$$\SUM{\infty}{(-1)^{n+1}b_n} = b_1 -b_2+b_3-b_4+\cdots$$

or $$\SUM{\infty}{(-1)^n b_n} = -b_1+b_2-b_3+b_4-\cdots$$

where $b_n\ge 0$ for all positive integers $n$.

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Theorem (Alternating Series Test)

An alternating series of the form $$\SUM{\infty}{(-1)^{n+1}b_n}\text{ or } \SUM{\infty}{(-1)^n b_n}$$ converges if

1. $0\le b_{n+1} \le b_n$ for all $n\ge 1$ and
2. $\lim_{n\to \infty}b_n=0$.

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Theorem (Remainders in Alternating Series)

Consider the alternating series of the form $$\SUM{\infty}{(-1)^{n+1}b_n} \text{ or }\SUM{\infty}{(-1)^nb_n}$$ that satisfies the hypotheses of the alternating series test.

Let $S$ denote the sum of the series and $S_N$ denote the $N^{\text{th}}$ partial sum. For any integer $N\ge 1$, the remainder $R_N = S - S_N$ satisfies $$|R_N|\le b_{N+1}$$

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Definition (Absolute and Condition Convergence)

A series $\SUM{\infty}{a_n}$ exhibits absolute convergence if $\SUM{\infty}{|a_n|}$ converges. A series $\SUM{\infty}{a_n}$ exhibits conditional convergence if $\SUM{\infty}{a_n}$ converges but $\SUM{\infty}{|a_n|}$ diverges.

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Theorem (Absolute Convergence Implies Convergence)

If $\SUM{\infty}{|a_n|}$ converges, then $\SUM{\infty}{a_n}$ converges.

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Ratio and Root Tests

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Theorem (Ratio Test)

Let $\SUM{\infty}{a_n}$ be a series with nonzero terms. Let $$\rho = \lim_{n\to \infty}\left|\frac{a_{n+1}}{a_n}\right|.$$

1. If $0\le\rho <1$, then $\SUM{\infty}{a_n}$ converges absolutely.
2. If $\rho >1$ or $\rho = \infty$, then $\SUM{\infty}{a_n}$ diverges.
3. If $\rho = 1$, the test does not provide any information.

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Theorem (Root Test)

Consider the series $\SUM{\infty}{a_n}$. Let $$\rho = \lim_{n\to \infty}\sqrt[n]{\left|a_n\right|}$$

1. If $0\le \rho <1$, then $\SUM{\infty}{a_n}$ converges absolutely.
2. If $\rho > 1$ or $\rho = \infty$, then $\SUM{\infty}{a_n}$ diverges.
3. If $\rho = 1$, the test does not provide any information.

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Series or Test	Conclusions	Comments
Divergence Test For any series $\sum_{n=1}^{\infty}(a_{n})$, evaluate $\lim_{n\to\infty}(a_{n})$.	If $\lim_{n\to\infty}(a_{n})=0$, the test is inconclusive.	The test cannot prove convergence of a series.
	If $\lim_{n\to\infty}(a_{n})\ne0$, the series diverges.
Geometric Series $\sum_{n=1}^{\infty}\left(ar^{n-1}\right)$	If $|r|<1$, the series converges to $\frac{a}{1-r}$.	Any geometric series can be re-indexed to be written in the form $a+ar+ar^{2}+\cdots$, where $a$ is the initial term and $r$ is the ratio.
	If $|r|\ge1$, the series diverges.
$p$-Series $\sum_{n=1}^{\infty}\left(\frac{1}{n^{p}}\right)$	If $p>1$, the series converges.	For $p=1$, we have the harmonic series $\sum_{n=1}^{\infty}\left(\frac{1}{n}\right)$.
	If $p\le1$, the series diverges.
Comparison Test For $\sum_{n=1}^{\infty}(a_{n})$ with nonnegative terms, compare with a known series $\sum_{n=1}^{\infty}(b_{n})$.	If $a_{n}\le b_{n}$ for all $n\ge N$ and $\sum_{n=1}^{\infty}(b_{n})$ converges, then $\sum_{n=1}^{\infty}(a_{n})$ converges	Typically used for a series similar to a geometric or $p$-series. It can sometimes be difficult to find an appropriate series.
	If $a_{n}\ge b_{n}$ for all $n\ge N$ and $\sum_{n=1}^{\infty}(b_{n})$ diverges, then $\sum_{n=1}^{\infty}(a_{n})$ diverges.
Limit Comparison Test For $\sum_{n=1}^{\infty}(a_{n})$ with positive terms, compare with a series $\sum_{n=1}^{\infty}(b_{n})$ by evaluating $L=\lim_{n\to\infty}\frac{a_{n}}{b_{n}}$.	If $L$ is a real number and $L\ne0$, then $\sum_{n=1}^{\infty}(a_{n})$ and $\sum_{n=1}^{\infty}(b_{n})$ both converge or both diverge.	Typically used for a series similar to a geometric or p-series. Often easier to apply than the comparison test.
	If $L=0$ and $\sum_{n=1}^{\infty}(b_{n})$ converges, then $\sum_{n=1}^{\infty}(a_{n})$ converges.
	If $L=\infty$ and $\sum_{n=1}^{\infty}(b_{n})$ diverges, then $\sum_{n=1}^{\infty}(a_{n})$ diverges.
Integral Test If there exists a positive, continuous, decreasing function $f$ such that $a_{n}=f(n)$ for all $n\ge N,$evaluate $\int_{N}^{\infty}f(x)\,dx$.	$\int_{N}^{\infty}f(x)\,dx$ and $\sum_{n=1}^{\infty}(a_{n})$ both converge or both diverge.	Limited to those series for which the corresponding function $f$ can be easily integrated.
Alternating Series $\sum_{n=1}^{\infty}\left((-1)^{n+1}b_{n}\right)$ or $\sum_{n=1}^{\infty}\left((-1)^{n}b_{n}\right)$	If $b_{n+1}\le b_{n}$ for all $n\ge1$ and $b_{n}\to0$ as $n\to\infty$, then the series converges.	Only applies to alternating series.
Ratio Test For any series $\sum_{n=1}^{\infty}(a_{n})$ with nonzero terms, let $\rho=\lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_{n}}\right|$.	If $0\le\rho<1$, the series converges absolutely.	Often used for series involving factorials or exponential.
	If $\rho>1$ or $\rho=\infty$, the series diverges.
	If $\rho=1$, the test is inconclusive.
Root Test For any series $\sum_{n=1}^{\infty}(a_{n})$ with nonzero terms, let $\rho=\lim_{n\to\infty}\sqrt[n]{|a_{n}|}$.	If $0\le\rho<1$, the series converges absolutely.	Often used for series where $|a_n|=b_n^n$
	If $\rho>1$ or $\rho=\infty$, the series diverges.
	If $\rho=1$, the test is inconclusive.