Limit of a Series

Definition

$\underset{n\to \infty }{\lim }{S}_n=S\Rightarrow \sum _{n=1}^{\infty }{a}_n=S$

A Series $\sum _{n=1}^{\infty }{a}_n$ is said to be convergent when the Sequence of partial sums $(S_n)$ has a finite limit $S\in \mathbb{R}$. Otherwise, the series is said to be divergent.

Properties

Absolute Convergence

Absolute Convergence

Definition

A Series $\sum _{n=1}^{\infty }{a}_n$ converges absolutely if the series of absolute values $\sum _{n=1}^{\infty }|a_n|$ converges.

Link to original

Conditional Convergence

A Series $\sum _{n=1}^{\infty }{a}_n$ is conditionally convergent if it is convergent but not absolutely convergent.

Linearity

Suppose $\alpha ,\beta ,\lambda ,\mu \in \mathbb{R}$, and $\sum _{n=1}^{\infty }{a}_n=\alpha ,\sum _{n=1}^{\infty }{b}_n=\beta$ Then, $\sum _{n=1}^{\infty }(\lambda a_n\pm \mu b_n)=\lambda \alpha \pm \mu \beta$

Facts

If $\sum _{n=1}^{\infty }{a}_n$ converges, then $\underset{n\to \infty }{\lim }{a}_n=0$

Suppose $\sum _{n=1}^{\infty }{a}_n$ is a series, and $\sum _{n=1}^{\infty }{a}_{f(n)}$ is its rearranged series. $\sum _{n=1}^{\infty }{a}_n=\sum _{n=1}^{\infty }|a_n|=L\Rightarrow \forall f,\sum _{n=1}^{\infty }{a}_{f(n)}=L$

When a Series is absolutely convergent, any rearranged series will still converge to the same value.