# Infinite Series
As opposed to [[Series|finite series]], infinite series is the **sum of all terms of an infinite [[Sequence|sequence]]. It is said to exist only if the [[Limit]] of its partial sums exists, as the number of elements approaches infinity. In that case, it is called a *convergent* series. Otherwise it is called a *divergent* series.**
$
\sum_{j=0}^\infty a_j \text { exists if } \lim_{n \to \infty} \sum_{j=0}^n a_j \text { exists}
$
Note that because we're determining the sum based on sums of subsequences with increasing number of elements, rearranging terms is not allowed.
**If two infinite series $A$ and $B$ are both convergent, operations between the two can be done term by term** i.e. $A[0] \circ B[0]$, $A[1] \circ B[1]$, etc. and one series can even be shifted to the right.
**The distributive also works for convergent series**, so if you create a series by multiplying each term of an initial series by a constant, the new series' sum will also be a multiple of the constant and the initial series' sum.
**For divergent series, shifting, distribution and term-by-term operations does not lead to consistent summation.**
## Convergence Tests
Because partial sums need to converge, if $\lim_{i \to \infty} |a_j| > 0$, the sum must be divergent. But $\lim_{j \to \infty} a_j = 0$ alone is not enough to guarantee the infinite sum is convergent (e.g. $\sum_{j=1}^\infty \frac 1 j$).
[[The Ratio Test]], and [[The Root Test]] can be used to easily test if an infinite sum is convergent.
## Well Known Infinite Series
### Geometric Infinite Series
![[Geometric Infinite Series#^45eu81]]
### Arithmetic-Geometric Infinite Series
![[Arithmetic-Geometric Infinite Series#^a597aa]]
### The Exponential Function and Trigonometric Functions
[[The Exponential Function]] is the most important infinite series because it yields the definition of $e$.
![[The Exponential Function#^fo2wdr]]
Using [[Euler's Formula]], the formulas of [[Trigonometric Functions]] can be defined using infinite series as well.
![[Expressing Trigonometric Functions Using The Exponential Function#^s1jedx]]
### Taylor Series
![[Taylor Series#^qnnyu9]]
## Rules
- [[Cauchy Product]]