Series Convergence

Definition

A series $\sum a_n$ converges if:

• The series of Partial Sums $\{S_n\}$ converges
• Or, $\exists s\in \mathbb{R},{\lim }_{n\to \infty }{S}_n=s$

Proving Series

• By definition
• Integral Test
• Div Test
• GS Test
• P-Series Test
• Comparison Theorem for Series
• Liebniz Series Test
• Ratio Test

Properties

Additive Property

1. If $\sum a_n$ converges to $c$
2. If $\sum b_n$ converges to $s$
3. Then. $\sum (a_n+b_n)$ converges to $c+s$

Constant Multiple Property

1. If $\sum a_n$ converges to $c$
2. Then, $\sum k{a}_n$ converges to $kc$

Vanishing Condition

1. If $\sum a_n$ converges to $c$
2. Then, as we approach the infinite-th term of the set, ${\lim }_{n\to \infty }{a}_n=0$ Proof of Series Vanishing Condition

Intuition

• In order to have a sum that is a constant, all the elements of the set must be infinitesimally small

Example

Example 1

Does ${\sum }_{n=1}^{\infty }\ln \left(\frac{n+1}{n}\right)$ converge or diverge?

1. Note that ${\sum }_{n=1}^{\infty }\ln \left(\frac{n+1}{n}\right)={\sum }_{n=1}^{\infty }[\ln (n+1)-\ln (n)]$
   1. Note, $S_n=a_1+a_2+\cdots +a_{n-1}+a_n$
   2. $=\ln (n+1)-\ln (1)$ by Telescoping Series
   3. $=\ln (n+1)$
2. Then, ${\lim }_{n\to \infty }{S}_n=\ln (\infty )=\infty$, and as $\infty$ is NaN, then this limit DNE