Series

Infinite series

An infinite series (or just series) of constants $a_k$ is defined as a limit of finite series: ${\sum }_{k=0}^{\infty }{a}_k={\lim }_{n\to \infty }{\sum }_{k=0}^n{a}_k$

Geometric Series

A geometric series can be written in the form ${\sum }_{k=0}^{\infty }a{r}^k=a+ar+a{r}^2+a{r}^3+\dots$ The sum of Geometric series is given by $S_n=\frac{a(1-r^{n+1})}{1-r}$

new kind of series from CS370

Doesn’t have to sum to infinity ${\sum }_{j=0}^{N-1}{x}^j=\frac{x^N-1}{x-1}$ where $x\ne 1$

Telescoping Series

a telescoping series is a series whose general term can be written as the difference of two consecutive terms of a sequence, i.e. $t_n=a_n-a_{n+1}$

Convergence / Divergence Tests

title: Convergence Test
Used on geometric series.
$$\sum_{k=0}^\infty{ar^k} = \frac{a}{1-r} \quad \text{if  } |r| < 1 \text{, and is divergent otherwise.}$$

title: $n^{th}$ Term Test (Test for Divergence)
If $\lim_{k \to \infty} a_k \neq 0$, then $\sum{a_k}$ diverges.
 
*DANGER: If $\lim_{n \to \infty} a_n = 0$, we cannot make any conclusions about divergence or convergence.*

title: The Integral Test (Condition to use: all $a_k > 0$)
**Theorem**: Consider a series $\sum_{k=k_0}^\infty{a_k}$. Let $f$ be a function which is **continuous, positive, and decreasing** on $(k_0,\infty)$, with $f(k) = a_k$ for all $k \geq k_0$.
 
1. If $\int_{k_0}^{\infty}f(x)dx$ converges, then $\sum_{k=k_0}^\infty{a_k}$ converges.
2. If $\int_{k_0}^{\infty}f(x)dx$ diverges, then $\sum_{k=k_0}^\infty{a_k}$ diverges.

title: Convergence of P-Series
$$\sum \frac{1}{k^p} \text{converges if } p > 1, \text{ and diverges if } p 
\leq 1$$

title:Direct Comparison Test (Condition to use: $a_k > 0$)
Given a positive series $\sum a_k$. If we find another series $\sum b_k$, where
1. $a_k \leq b_k$, and if $\sum b_k$ converges, then $\sum a_k$ converges.
1. $a_k \geq b_k$, and if $\sum b_k$ diverges, then $\sum a_k$ diverges.

title: Limit Comparison Test ($a_k > 0$)
Assume that the following limit exists (or $= \infty$):
$$L = \lim_{k \to \infty} \frac{a_k}{b_k}$$
 
- If $L = C$ (non-zero constant), then $\sum a_k$ converges $\iff \sum b_k$ converges 
	- They either both converge or both diverge
- If $L = 0$, and $\sum b_k$ converges, then $\sum a_k$ also converges
- If $L = \infty$, and $\sum b_k$ diverges, then $\sum a_k$ also diverges

title: Alternating Series Test (AST) / Leibniz Test
Consider the series $\sum_{k=0}^\infty{(-1)^k a_k}$, where $a_k > 0$ for every $k$. If
1. ${a_k}$ is a decreasing sequence, and 
2. $\lim_{k \to \infty} a_k = 0$,
 
then the series converges.

title: Alternative Series Estimation Theorem (ASET)
Consider the series $\sum_{k=0}^\infty{(-1)^k a_k}$, where $a_k > 0$ for every $k$. If
1. ${a_k}$ is a decreasing sequence, and 
2. $\lim_{k \to \infty} a_k = 0$,
 
Then $|s-s_n| \leq a_{n+1}$.

title: [[The Ratio Test]]
Assume that the following limit exists (or $= \infty$):
$$L = \lim_{k \to \infty} \Big\vert \frac{a_{k+1}}{a_k}\Big\vert$$
 
- If $L < 1$, then $\sum a_k$ is absolutely convergent.
- If $L > 1$, then $\sum a_k$ is divergent.
- If $L = 1$, then the test fails.

title: The Root Test
Assume that the following limit exists (or $= \infty$):
$$L = \lim_{k \to \infty} \sqrt[k]{|a_k|} $$
 
- If $L < 1$, then $\sum a_k$ is absolutely convergent.
- If $L > 1$, then $\sum a_k$ is divergent.
- If $L = 1$, then the test fails.

Conditional vs. Absolute Convergence

A series $\sum a_k$ is called absolutely convergent if $\sum |a_k|$ converges.

title: Absolute Convergence Implies Convergence 
If the series $\sum |a_k|$ converges, then $\sum a_k$ also converges.

A series $\sum a_k$ is said to be conditionally convergent if it converges, but the series $\sum |a_k|$ diverges.

To check between for absolute convergence, we use The Ratio Test.

Related

• Power Series