# Convergence of Series
When analyzing the convergence or divergence of a [[Sequences|sequence]], we only need to take the limit of its function and identify whether or not it exists.
The process is more complicated for [[Sums and Series|series]] as the elements are being added together. Several **convergence tests** exist as methods for testing the nature of convergence for a given series.
## Types of Convergence
| Type | Description |
| ------------------------ | ----------- |
| Convergent | |
| Absolutely Convergent | |
| Conditionally Convergent | |
| Divergent | |
If the series of absolute values $\sum|a_n|$ converges then the original series $\sum a_n$ converges as well.
In this case, the original series $\sum a_n$ is said to be **absolutely convergent**.
- Every absolute convergent series
The series $\sum a_n$ is **conditionally convergent** if it converges, but the series of absolute values $\sum |a_n|$ *diverges*.
### Term Rearrangement
> See also:
> - [[Mathematical Operations]]
When working with a finite number of terms, the standard property of addition/subtraction
![[Pasted image 20240328083925.png|400]]
## Convergence Tests
> [!summary]- Summary of Convergence Tests
>
> ![[screencapture-knewton-learn-course-fe14cad5-043c-4e35-b9bf-301ea0ebf937-assignment-409034f6-f21c-4862-bdf7-4e6e5b8bd852-2025-07-03-11_19_31 (1).png]]
>
### Limit Comparison Test
Let $\sum a_n$
### The nth Term Test
### The Integral Test
> See also:
> - [[Integration#Improper Integrals|Improper Integrals]]
Let $\{a_n\}$ be a sequence of positive terms.
Suppose that there is a positive decreasing function $f(n)$, such that $f(n)=a_n$.
Then the summation of $a_n$ and the integral of $f(x)$ converge or diverge together.
In other words:
- If $\int^{\infty}_{1} f(n)dn$ is convergent, then $\sum f(n)$ is also convergent.
- If $\int^{\infty}_{1} f(n)dn$ is divergent, then $\sum f(n)$ is also divergent.
### The p-Series Test
Suppose $p \neq 1$:
Suppose $p=1$:
### The Ratio Test
The ratio test aims to determine potential convergence by calculating the ratio of consecutive terms ($a_{n+1}$ vs $a_{n}$). In other words, *the growth rate of the series*.
$p=\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$
- If $p<1$ then the series *converges*
- If $p>1$ then the series *diverges*
- If $p=1$ then the test is *inconclusive*
identifying if the terms of the series eventually behave geometrically
![[Pasted image 20240328080518.png|400]]
- Try to use the ratio test when working with factorials to help cancel out terms
### The Root Test
$p=\lim_{n \to \infty} \sqrt[n]{|a_n|}$
- If $p<1$ then the series *converges absolutely*
- If $p>1$ then the series *diverges*
- If $p=1$ then the test is inconclusive
![[Pasted image 20240328081338.png|400]]
### The Alternating Series Test
$\sum(-1)^{n+1} u_n$
This series converges if the following conditions are satisfied:
1. The $u_n$’s are all positive.
2. The $u_n$’s are eventually nonincreasing ($u_n \ge u_{n+1}$ for all $n \ge N$, for some integer $N$)
3. $u_n$ → 0
![[Pasted image 20240410071132.png|300]]
![[Pasted image 20240328081839.png|500]]
![[Pasted image 20240328082142.png|500]]
![[Pasted image 20240328082425.png|500]]
![[Pasted image 20240328082324.png|500]]
#### Alternating Series Estimation Theorem
- Apply this theorem to approximate the sum of an alternating series
- Establish a bound for the error in our approximation
our limiting value will always be between or equal to the distance between two consecutive terms
![[Pasted image 20240328083035.png|400]]
![[Pasted image 20240328083309.png|400]]