### Key Concepts

### 5.1 Sequences

- To determine the convergence of a sequence given by an explicit formula ${a}_{n}=f(n),$ we use the properties of limits for functions.
- If $\left\{{a}_{n}\right\}$ and $\left\{{b}_{n}\right\}$ are convergent sequences that converge to $A$ and $B,$ respectively, and $c$ is any real number, then the sequence $\left\{c{a}_{n}\right\}$ converges to $c\xb7A,$ the sequences $\{{a}_{n}\pm {b}_{n}\}$ converge to $A\pm B,$ the sequence $\{{a}_{n}\xb7{b}_{n}\}$ converges to $A\xb7B,$ and the sequence $\left\{{a}_{n}\text{/}{b}_{n}\right\}$ converges to $A\text{/}B,$ provided $B\ne 0.$
- 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 $\left\{{r}^{n}\right\}$ converges if and only if $\left|r\right|<1$ or $r=1.$

### 5.2 Infinite Series

- Given the infinite series

$$\sum}_{n=1}^{\infty}{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\text{\cdots$$

and the corresponding sequence of partial sums $\left\{{S}_{k}\right\}$ where

$${S}_{k}={\displaystyle \sum}_{n=1}^{k}{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\text{\cdots}+{a}_{k},$$

the series converges if and only if the sequence $\left\{{S}_{k}\right\}$ converges. - The geometric series $\sum}_{n=1}^{\infty}a{r}^{n-1$ converges if $\left|r\right|<1$ and diverges if $\left|r\right|\ge 1.$ For $\left|r\right|<1,$

$$\sum}_{n=1}^{\infty}a{r}^{n-1}=\frac{a}{1-r}.$$ - The harmonic series

$$\sum}_{n=1}^{\infty}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\text{\cdots$$

diverges. - A series of the form $\sum _{n=1}^{\infty}[{b}_{n}-{b}_{n+1}]}=[{b}_{1}-{b}_{2}]+[{b}_{2}-{b}_{3}]+[{b}_{3}-{b}_{4}]+\text{\cdots}+[{b}_{n}-{b}_{n+1}]+\text{\cdots$

is a telescoping series. The $k\text{th}$ partial sum of this series is given by ${S}_{k}={b}_{1}-{b}_{k+1}.$ The series will converge if and only if $\underset{k\to \infty}{\text{lim}}{b}_{k+1}$ exists. In that case,

$$\sum _{n=1}^{\infty}[{b}_{n}-{b}_{n+1}]}={b}_{1}-\underset{k\to \infty}{\text{lim}}\left({b}_{k+1}\right).$$

### 5.3 The Divergence and Integral Tests

- If $\underset{n\to \infty}{\text{lim}}{a}_{n}\ne 0,$ then the series $\sum}_{n=1}^{\infty}{a}_{n$ diverges.
- If $\underset{n\to \infty}{\text{lim}}{a}_{n}=0,$ the series $\sum}_{n=1}^{\infty}{a}_{n$ may converge or diverge.
- If $\sum}_{n=1}^{\infty}{a}_{n$ is a series with positive terms ${a}_{n}$ and $f$ is a continuous, decreasing function such that $f\left(n\right)={a}_{n}$ for all positive integers $n,$ then

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

either both converge or both diverge. Furthermore, if $\sum}_{n=1}^{\infty}{a}_{n$ converges, then the $N\text{th}$ partial sum approximation ${S}_{N}$ is accurate up to an error ${R}_{N}$ where ${\int}_{N+1}^{\infty}f\left(x\right)dx<{R}_{N}<{\displaystyle {\int}_{N}^{\infty}f\left(x\right)dx}}.$ - The
*p*-series $\sum}_{n=1}^{\infty}1\text{/}{n}^{p$ converges if $p>1$ and diverges if $p\le 1.$

### 5.4 Comparison Tests

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

### 5.5 Alternating Series

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

### 5.6 Ratio and Root Tests

- For the ratio test, we consider

$$\rho =\underset{n\to \infty}{\text{lim}}\left|\frac{{a}_{n+1}}{{a}_{n}}\right|.$$

If $\rho <1,$ the series $\sum _{n=1}^{\infty}{a}_{n}$ converges absolutely. If $\rho >1,$ the series diverges. If $\rho =1,$ the test does not provide any information. This test is useful for series whose terms involve factorials. - For the root test, we consider

$$\rho =\underset{n\to \infty}{\text{lim}}\sqrt[n]{\left|{a}_{n}\right|}.$$

If $\rho <1,$ the series $\sum _{n=1}^{\infty}{a}_{n}$ converges absolutely. If $\rho >1,$ the series diverges. If $\rho =1,$ 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 $p-\text{series,}$ consider one of the comparison tests.