# 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]]