# Sums and Series
A **series** is the *sum* of the terms within a [[sequences|sequence]].
## Summation Notation
**Summation** is the process of *adding things together* and is a common notation used to represent mathematical series.
It is represented mathematically by the Greek letter sigma: $\sum$
> [!abstract] **Sigma/Summation Notation**
>
> $\sum^{n}_{i=1} a_i = a_1 + a_2 + a_3 + ... a_{n-1} + a_n$
>
> - The variable $i$ is called the *index* and takes *only integer* values
> - The index $i$ starts at $i=1$ and ends at $i=n$
> - $n$ is the *end value* and represents the *# of elements*
> - $a_i$ represents the formula for the $i$-th term
The sigma notation above would be read as "the sum as $i$ goes from $1$ to $n$ of $a_i
quot;.
The final sum of the *sequence* ($a_1 + a_2 + ... a_n$) is known as the **series** (sometimes used interchangeably with sigma).
## Properties of Summation
> [!rule]+ Properties of Sigma Notation
> 1. Let $a_1, a_2, ..., a_n$ and $b_1, b_2, ..., b_n$ represent two sequences of terms.
> 2. Let $c$ be a constant.
> 3. The following properties hold for all positive integers $n$ for integers $m$, with $1 \ge m \ge n$:
>
> **The Constant Value Rule:**
> $\sum^{n}_{i=1} c = nc$
> When $i \ne 1$:
> $\sum^{a}_{i=b} c = c(a-b+1)$
>
> ---
> **The Sum/Difference Rule:**
> $\sum^{n}_{i=1} (a_i \pm b_i) = \sum^{n}_{i=1} a_i \pm \sum^{n}_{i=1} b_i$
>
> ---
> **The Constant Multiple Rule:**
> $\sum^{n}_{i=1} ca_i = c \cdot \sum^{n}_{i=1} a_i$
>
> ---
> **The Midpoint Index Rule**
> $\sum^{n}_{i=1} a_i = \sum^{m}_{i=1} a_i + \sum^{n}_{i=m+1} a_i$
> [!rule]+ Sums and Powers of Integers
> 1. The sum of $n$ integers is given by:
> $\sum^{n}_{i=1} i = 1 + 2 + ... + n = \frac{n(n+1)}{2}$
> 2. The sum of consecutive integers squared is given by:
>$\sum^{n}_{i=1} i^2 = 1^2 + 2^2 + ... + n^2 = \frac{n(n+1)(2n+1)}{6}$
> 3. The sum of consecutive integers cubed is given by:
> $\sum^{n}_{i=1} i^3 = 1^3 + 2^3 + ... + n^3 = \frac{n^2(n+1)^2}{4}$
![[Pasted image 20230305224314.png]]
## Notable Series
> See also:
> - [[Sequences#Notable Sequences]]
- [ ] Geometric
- [ ] Alternating Series
- [ ] Power Series
https://web.ma.utexas.edu/users/m408s/2016/LM11-5-8.html
### Power Series
A power series expresses a function as an infinite polynomial.
$\sum^{\infty}_{n=0}c_n \cdot x^n$
$f(x)=c_0 + c_1x + c_2x^2 + c_3x^3 + ...$
- $c_n$ are the coefficients of the series
The domain of this function is the set of $x$ for which $f(x)$ converges
The more terms that are included
---
For a given power series $\sum^{\infty}_{n=0}c_n(x-a)^n$ there are only three possibilities:
1. The series converges only when $x=a$
2. The series converges for all $x$ ($-\infty \to \infty$)
3. There is a positive number $R$ such that the series:
- Converges if $|x-a|<R$
- Diverges if $|x-a|>R$
![[Pasted image 20240410075954.png|400]]
The radius of convergence can be determined using [[Convergence of Series#The Ratio Test|the ratio test]]
### Convergence Theorem for Power Series
## Taylor & Maclaurin Series