Series

Infinite series

An infinite series (or just series) of constants is defined as a limit of finite series:

Geometric Series

A geometric series can be written in the form The sum of Geometric series is given by

new kind of series from CS370

Doesn’t have to sum to infinity where

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.

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 is called absolutely convergent if converges.

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

A series is said to be conditionally convergent if it converges, but the series diverges.

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