- 9 | Sequences and Series
- 10 | Approximating Functions Using Series
- 12 | Functions of Several Variables
- 13 | A Fundamental Tool: Vectors
- MA 123: The SandBox
- MA 123: Homework
- MA 123: Workshop
- List of Maclaurin Series of Some Common Functions
- Review Problems

A series which can be written in the form

$$

\sum_{n = 1}^{\infty} \frac{1}{n}

$$

is called a

$$

\sum_{n = 1}^{\infty} \frac{1}{n}

$$

is called a

**harmonic series**and diverges.A

$$

\sum_{n = 1}^{\infty} ar^{n-1} = a + ar + ar^2 + \cdots

$$

and

$$

\sum_{n =1}^{\infty} ar^{n-1} =

\begin{cases}

\frac{a}{1-r}, & \; |r| <1 \\ \\ \text{divergent}, & \; \text{otherwise} \end{cases} $$

**geometric series**is a series of the form$$

\sum_{n = 1}^{\infty} ar^{n-1} = a + ar + ar^2 + \cdots

$$

and

$$

\sum_{n =1}^{\infty} ar^{n-1} =

\begin{cases}

\frac{a}{1-r}, & \; |r| <1 \\ \\ \text{divergent}, & \; \text{otherwise} \end{cases} $$

A series which can be written in the form

$$

\sum_{n=1}^{\infty} \frac{1}{n^p}

$$

is called a

$$

\sum_{n=1}^{\infty} \frac{1}{n^p} =

\begin{cases}

\text{convergent}, & \; \text{ if } p > 1 \\

\text{divergent}, & \; \text{ if } p \leq 1

\end{cases}

$$

$$

\sum_{n=1}^{\infty} \frac{1}{n^p}

$$

is called a

**\(p\)-series**and$$

\sum_{n=1}^{\infty} \frac{1}{n^p} =

\begin{cases}

\text{convergent}, & \; \text{ if } p > 1 \\

\text{divergent}, & \; \text{ if } p \leq 1

\end{cases}

$$

**The Divergence Test:**If \(\displaystyle \lim_{n \rightarrow \infty} a_n \) does not exist or \(\displaystyle \lim_{n \rightarrow \infty} a_n \not = 0\), then the series \(\displaystyle \sum_{n = 1}^{\infty} a_n\) is divergent.

**The Alternating Series Test:**Given a series of the form

$$

\sum (-1)^n a_n

$$

with \(a_n >0\), if

- \( \displaystyle \lim_{n \rightarrow \infty} a_n = 0 \)
- \(a_n\) is decreasing. \((a_{n+1} \leq a_n)\)

then the series converges.

**The Integral Test:**Let \(f(x)\) be a positive, continuous, decreasing function on \([1, \infty) \) with \(f(n) = a_n\) for \(n \geq 1\). Then

$$

\sum_{n = 1}^{\infty} a_n \text{ and } \int_{1}^{\infty} f(x) \; dx

$$

either both converge or both diverge.

**The (Direct) Comparison Test:**Given a series \(\sum a_n\) with \(a_n >0\) choose a comparison series \(\sum b_n\) with \(b_n >0\).

- If \(a_n \leq b_n\) and \(\sum b_n\) converges, then so does \(\sum a_n\).
- If \(b_n \leq a_n\) and \(\sum b_n\) diverges, then so does \(\sum a_n\).

**The Limit Comparison Test:**Given a series \(\sum a_n\) with \(a_n >0\) choose a comparison series \(\sum b_n\) with \(b_n >0\).

If

$$

\lim_{n \rightarrow \infty} \frac{a_n}{b_n}

$$

is a positive number then either both series converge or both diverge. Otherwise, the test is inconclusive.

A series \(\sum a_n\) is

**absolutely convergent**provided \(\sum |a_n|\) converges. That is, a series converges absolutely if the same series with absolute values around the terms of the sum converges.If a series converges absolutely, then it converges.

**Ratio Test:**Given \(\sum a_n\) with \(a_n \not = 0 \) then

- If \(\displaystyle \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n}\right|<1\), the series converges absolutely.
- If \(\displaystyle \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n}\right|>1\), the series diverges.
- If \(\displaystyle \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n}\right|=1\), the test is inconclusive.

**Root Test:**Given \(\sum a_n\) then

- If \(\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{\left| a_n\right|}<1\), the series converges absolutely.
- If \(\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{\left| a_n\right|}>1\), the series diverges.
- If \(\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{\left| a_n\right|}=1\), the test is inconclusive.