The Definition of Infinite Series

Basic Definition

An infinite sequence is a sequence determined by an explicit formula or a recursion formula.

The infinite series is the sum of an infinite sequence, denoted as ${\displaystyle S_n=\sum _{n=1}^{\mathrm{\infty }}{a}_n}$.

Convergence and Divergence

For Infinite Sequence

Similar to improper integrals, the convergence and divergence of an infinite sequence is defined by its limit when ${\displaystyle n}$ is approaching infinity.

When a sequence ${\displaystyle a_n}$ is converge to ${\displaystyle L}$, we say:

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=L}$

We can use a method like the ${\displaystyle ϵ-\delta }$ definition to give this a rigorous proof:

If for ${\displaystyle n\ge N}$ with a given ${\displaystyle N>0}$, ${\displaystyle |a_n-L|<ϵ}$ is true for each given ${\displaystyle ϵ}$(no matter how small it is), we say the sequence is converge to ${\displaystyle L}$.

If the limit does not exists, the infinite sequence is diverge.

For Infinite Series

If ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\sum _{n=1}^{\mathrm{\infty }}{a}_n=L}$, we say it is converge.

Theorems

Squeeze Theorem

If ${\displaystyle a_n,c_n}$ both converge to ${\displaystyle L}$ when ${\displaystyle a_n<b_n<c_n}$, then ${\displaystyle b_n}$ also converges to ${\displaystyle L}$.

Absolute Value

If ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}|a_n|=0}$, then ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=0}$.

${\displaystyle n}$th Term Test for Divergence

If the infinite series ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}{a}_n}$ converges, then ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n=0}$. Equivalently, if ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{a}_n\ne 0}$ or does not exist, the series must diverge.

Notice that this theorem can only be used to prove divergence but not convergence!