Essential Concepts

Sequences and Their Properties

• To determine the convergence of a sequence given by an explicit formula [latex]{a}_{n}=f\left(n\right)[/latex], we use the properties of limits for functions.
• If [latex]\left\{{a}_{n}\right\}[/latex] and [latex]\left\{{b}_{n}\right\}[/latex] are convergent sequences that converge to [latex]A[/latex] and [latex]B[/latex], respectively, and [latex]c[/latex] is any real number, then the sequence [latex]\left\{c{a}_{n}\right\}[/latex] converges to [latex]c\cdot A[/latex], the sequences [latex]\left\{{a}_{n}\pm {b}_{n}\right\}[/latex] converge to [latex]A\pm B[/latex], the sequence [latex]\left\{{a}_{n}\cdot {b}_{n}\right\}[/latex] converges to [latex]A\cdot B[/latex], and the sequence [latex]\left\{\frac{{a}_{n}}{{b}_{n}}\right\}[/latex] converges to [latex]\frac{A}{B}[/latex], provided [latex]B\ne 0[/latex].
• If a sequence is bounded and monotone, then it converges, but not all convergent sequences are monotone.
• If a sequence is unbounded, it diverges, but not all divergent sequences are unbounded.
• The geometric sequence [latex]\left\{{r}^{n}\right\}[/latex] converges if and only if [latex]|r|<1[/latex] or [latex]r=1[/latex].

Introduction to Series

• Given the infinite series
  
  [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\cdots[/latex]
  and the corresponding sequence of partial sums [latex]\left\{{S}_{k}\right\}[/latex] where
  
  [latex]{S}_{k}=\displaystyle\sum _{n=1}^{k}{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\cdots +{a}_{k}[/latex],
  the series converges if and only if the sequence [latex]\left\{{S}_{k}\right\}[/latex] converges.
• The geometric series [latex]\displaystyle\sum _{n=1}^{\infty }a{r}^{n - 1}[/latex] converges if [latex]|r|<1[/latex] and diverges if [latex]|r|\ge 1[/latex]. For [latex]|r|<1[/latex],
  
  [latex]\displaystyle\sum _{n=1}^{\infty }a{r}^{n - 1}=\frac{a}{1-r}[/latex].
• The harmonic series
  
  [latex]\displaystyle\sum _{n=1}^{\infty }\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots[/latex]
  diverges.
• A series of the form [latex]\displaystyle\sum _{n=1}^{\infty }\left[{b}_{n}-{b}_{n+1}\right]=\left[{b}_{1}-{b}_{2}\right]+\left[{b}_{2}-{b}_{3}\right]+\left[{b}_{3}-{b}_{4}\right]+\cdots +\left[{b}_{n}-{b}_{n+1}\right]+\cdots[/latex]
  is a telescoping series. The [latex]k\text{th}[/latex] partial sum of this series is given by [latex]{S}_{k}={b}_{1}-{b}_{k+1}[/latex]. The series will converge if and only if [latex]\underset{k\to \infty }{\text{lim}}{b}_{k+1}[/latex] exists. In that case,
  
  [latex]\displaystyle\sum _{n=1}^{\infty }\left[{b}_{n}-{b}_{n+1}\right]={b}_{1}-\underset{k\to \infty }{\text{lim}}\left({b}_{k+1}\right)[/latex].

The Divergence and Integral Tests

• If [latex]\underset{n\to \infty }{\text{lim}}{a}_{n}\ne 0[/latex], then the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] diverges.
• If [latex]\underset{n\to \infty }{\text{lim}}{a}_{n}=0[/latex], the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] may converge or diverge.
• If [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] is a series with positive terms [latex]{a}_{n}[/latex] and [latex]f[/latex] is a continuous, decreasing function such that [latex]f\left(n\right)={a}_{n}[/latex] for all positive integers [latex]n[/latex], then
  
  [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}\text{and}{\displaystyle\int }_{1}^{\infty }f\left(x\right)dx[/latex]
  either both converge or both diverge. Furthermore, if [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges, then the [latex]N\text{th}[/latex] partial sum approximation [latex]{S}_{N}[/latex] is accurate up to an error [latex]{R}_{N}[/latex] where [latex]{\displaystyle\int }_{N+1}^{\infty }f\left(x\right)dx<{R}_{N}<{\displaystyle\int }_{N}^{\infty }f\left(x\right)dx[/latex].
• The p-series [latex]\displaystyle\sum _{n=1}^{\infty }\frac{1}{{n}^{p}}[/latex] converges if [latex]p>1[/latex] and diverges if [latex]p\le 1[/latex].

Comparison Tests

• The comparison tests are used to determine convergence or divergence of series with positive terms.
• When using the comparison tests, a series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] is often compared to a geometric or p-series.

Alternating Series

• For an alternating series [latex]\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{b}_{n}[/latex], if [latex]{b}_{k+1}\le {b}_{k}[/latex] for all [latex]k[/latex] and [latex]{b}_{k}\to 0[/latex] as [latex]k\to \infty[/latex], the alternating series converges.
• If [latex]\displaystyle\sum _{n=1}^{\infty }|{a}_{n}|[/latex] converges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges.

Ratio and Root Tests

• For the ratio test, we consider
  
  [latex]\rho =\underset{n\to \infty }{\text{lim}}|\frac{{a}_{n+1}}{{a}_{n}}|[/latex].

  
  
  If [latex]\rho <1[/latex], the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges absolutely. If [latex]\rho >1[/latex], the series diverges. If [latex]\rho =1[/latex], the test does not provide any information. This test is useful for series whose terms involve factorials.

• For the root test, we consider
  
  [latex]\rho =\underset{n\to \infty }{\text{lim}}\sqrt[n]{|{a}_{n}|}[/latex].

  
  
  If [latex]\rho <1[/latex], the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges absolutely. If [latex]\rho >1[/latex], the series diverges. If [latex]\rho =1[/latex], the test does not provide any information. The root test is useful for series whose terms involve powers.

• For a series that is similar to a geometric series or [latex]p-\text{series,}[/latex] consider one of the comparison tests.

Key Equations

• Harmonic series
  
  [latex]\displaystyle\sum _{n=1}^{\infty }\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots[/latex]
• Sum of a geometric series
  
  [latex]\displaystyle\sum _{n=1}^{\infty }a{r}^{n - 1}=\frac{a}{1-r}\text{ for }|r|<1[/latex]
• Divergence test
  
  [latex]\text{If }{a}_{n}\nrightarrow 0\text{ as }n\to \infty ,\displaystyle\sum _{n=1}^{\infty }{a}_{n}\text{ diverges}[/latex].
• p-series
  
  [latex]{\displaystyle\sum _{n=1}^{\infty}} \dfrac{1}{n^{p}} \bigg\{ \begin{array}{l}\text{ converges if }p>1\\ \text{ diverges if }p\le 1\end{array}[/latex]
• Remainder estimate from the integral test
  
  [latex]{\displaystyle\int }_{N+1}^{\infty }f\left(x\right)dx<{R}_{N}<{\displaystyle\int }_{N}^{\infty }f\left(x\right)dx[/latex]
• Alternating series
  
  [latex]\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{b}_{n}={b}_{1}-{b}_{2}+{b}_{3}-{b}_{4}+\cdots \text{or}[/latex]
  
  [latex]\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n}{b}_{n}=\text{-}{b}_{1}+{b}_{2}-{b}_{3}+{b}_{4}-\cdots[/latex]

Glossary

absolute convergence

if the series [latex]\displaystyle\sum _{n=1}^{\infty }|{a}_{n}|[/latex] converges, the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] is said to converge absolutely

alternating series

a series of the form [latex]\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{b}_{n}[/latex] or [latex]\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n}{b}_{n}[/latex], where [latex]{b}_{n}\ge 0[/latex], is called an alternating series

alternating series test

for an alternating series of either form, if [latex]{b}_{n+1}\le {b}_{n}[/latex] for all integers [latex]n\ge 1[/latex] and [latex]{b}_{n}\to 0[/latex], then an alternating series converges

arithmetic sequence
a sequence in which the difference between every pair of consecutive terms is the same is called an arithmetic sequence

bounded above

a sequence [latex]\left\{{a}_{n}\right\}[/latex] is bounded above if there exists a constant [latex]M[/latex] such that [latex]{a}_{n}\le M[/latex] for all positive integers [latex]n[/latex]

bounded below

a sequence [latex]\left\{{a}_{n}\right\}[/latex] is bounded below if there exists a constant [latex]M[/latex] such that [latex]M\le {a}_{n}[/latex] for all positive integers [latex]n[/latex]

bounded sequence

a sequence [latex]\left\{{a}_{n}\right\}[/latex] is bounded if there exists a constant [latex]M[/latex] such that [latex]|{a}_{n}|\le M[/latex] for all positive integers [latex]n[/latex]

comparison test

if [latex]0\le {a}_{n}\le {b}_{n}[/latex] for all [latex]n\ge N[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] converges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges; if [latex]{a}_{n}\ge {b}_{n}\ge 0[/latex] for all [latex]n\ge N[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] diverges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] diverges

conditional convergence

if the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges, but the series [latex]\displaystyle\sum _{n=1}^{\infty }|{a}_{n}|[/latex] diverges, the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] is said to converge conditionally

convergence of a series
a series converges if the sequence of partial sums for that series converges

convergent sequence
a convergent sequence is a sequence [latex]\left\{{a}_{n}\right\}[/latex] for which there exists a real number [latex]L[/latex] such that [latex]{a}_{n}[/latex] is arbitrarily close to [latex]L[/latex] as long as [latex]n[/latex] is sufficiently large

divergence of a series

a series diverges if the sequence of partial sums for that series diverges

divergence test

if [latex]\underset{n\to \infty }{\text{lim}}{a}_{n}\ne 0[/latex], then the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] diverges

divergent sequence

a sequence that is not convergent is divergent

explicit formula

a sequence may be defined by an explicit formula such that [latex]{a}_{n}=f\left(n\right)[/latex]

geometric sequence

a sequence [latex]\left\{{a}_{n}\right\}[/latex] in which the ratio [latex]\frac{{a}_{n+1}}{{a}_{n}}[/latex] is the same for all positive integers [latex]n[/latex] is called a geometric sequence

geometric series

a geometric series is a series that can be written in the form

[latex]\displaystyle\sum _{n=1}^{\infty }a{r}^{n - 1}=a+ar+a{r}^{2}+a{r}^{3}+\cdots[/latex]

harmonic series

the harmonic series takes the form

[latex]\displaystyle\sum _{n=1}^{\infty }\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots[/latex]

index variable
the subscript used to define the terms in a sequence is called the index

infinite series

an infinite series is an expression of the form

[latex]{a}_{1}+{a}_{2}+{a}_{3}+\cdots =\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex]

integral test

for a series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] with positive terms [latex]{a}_{n}[/latex], if there exists a continuous, decreasing function [latex]f[/latex] such that [latex]f\left(n\right)={a}_{n}[/latex] for all positive integers [latex]n[/latex], then

[latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}\text{ and }{\displaystyle\int }_{1}^{\infty }f\left(x\right)dx[/latex]

limit comparison test

suppose [latex]{a}_{n},{b}_{n}\ge 0[/latex] for all [latex]n\ge 1[/latex]. If [latex]\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}\to L\ne 0[/latex], then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] both converge or both diverge; if [latex]\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}\to 0[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] converges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges. If [latex]\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}\to \infty[/latex], and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] diverges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] diverges

limit of a sequence
the real number [latex]L[/latex] to which a sequence converges is called the limit of the sequence

monotone sequence

an increasing or decreasing sequence

partial sum

the [latex]k\text{th}[/latex] partial sum of the infinite series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] is the finite sum

[latex]{S}_{k}=\displaystyle\sum _{n=1}^{k}{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\cdots +{a}_{k}[/latex]

p-series

a series of the form [latex]\displaystyle\sum _{n=1}^{\infty }\frac{1}{{n}^{p}}[/latex]

ratio test
for a series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] with nonzero terms, let [latex]\rho =\underset{n\to \infty }{\text{lim}}|\frac{{a}_{n+1}}{{a}_{n}}|[/latex]; if [latex]0\le \rho <1[/latex], the series converges absolutely; if [latex]\rho >1[/latex], the series diverges; if [latex]\rho =1[/latex], the test is inconclusive

recurrence relation

a recurrence relation is a relationship in which a term [latex]{a}_{n}[/latex] in a sequence is defined in terms of earlier terms in the sequence

remainder estimate

for a series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] with positive terms [latex]{a}_{n}[/latex] and a continuous, decreasing function [latex]f[/latex] such that [latex]f\left(n\right)={a}_{n}[/latex] for all positive integers [latex]n[/latex], the remainder [latex]{R}_{N}=\displaystyle\sum _{n=1}^{\infty }{a}_{n}-\displaystyle\sum _{n=1}^{N}{a}_{n}[/latex] satisfies the following estimate:

[latex]{\displaystyle\int }_{N+1}^{\infty }f\left(x\right)dx<{R}_{N}<{\displaystyle\int }_{N}^{\infty }f\left(x\right)dx[/latex]

root test
for a series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex], let [latex]\rho =\underset{n\to \infty }{\text{lim}}\sqrt[n]{|{a}_{n}|}[/latex]; if [latex]0\le \rho <1[/latex], the series converges absolutely; if [latex]\rho >1[/latex], the series diverges; if [latex]\rho =1[/latex], the test is inconclusive

sequence

an ordered list of numbers of the form [latex]{a}_{1},{a}_{2},{a}_{3}\text{,}\ldots[/latex] is a sequence

telescoping series

a telescoping series is one in which most of the terms cancel in each of the partial sums

term
the number [latex]{a}_{n}[/latex] in the sequence [latex]\left\{{a}_{n}\right\}[/latex] is called the [latex]n\text{th}[/latex] term of the sequence

unbounded sequence

a sequence that is not bounded is called unbounded