Infinite Series

An infinite series is an expression of the form

$\sum _{\mathit{n}=1}^{\infty }{\mathit{a}}_{\mathit{n}}={\mathit{a}}_1+{\mathit{a}}_2+{\mathit{a}}_3+...$

For each positive integer  k, the sum

${\mathit{S}}_{\mathit{k}}=$ $\sum _{\mathit{n}=1}^{\mathit{k}}{\mathit{a}}_{\mathit{n}}={\mathit{a}}_1+{\mathit{a}}_2+{\mathit{a}}_3+...+{\mathit{a}}_{\mathit{k}}$

is called the $\mathit{k}$th partial sum of the infinite series. The partial sums form a sequence $\left\{{\mathit{S}}_{\mathit{k}}\right\}$. 

If we can describe the convergence of a series to $\mathit{S}$, we call $\mathit{S}$ the sum of the series, and we write

$\sum _{\mathit{n}=1}^{\infty }{\mathit{a}}_{\mathit{n}}=\mathit{S}$

Convergent Series

If the sequence of partial sums converges to a real number $\mathit{S}$, the infinite series converges.

Example: $\sum _{\mathit{n}=1}^{\infty }\frac{1}{2^{\mathit{n}}}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...$

Divergent Series

If the sequence of partial sums diverges, we have the divergence of a series.

Example: $\sum _{\mathit{n}=1}^{\infty }\frac{1}{\mathit{n}}=1+\frac{1}{2}+\frac{1}{3}+...$

Example 1:

Determine whether the series ${\displaystyle \underset{\mathit{n}=1}{\overset{\infty }{ \sum }}\mathit{ }\left(\frac{\mathit{n}+5}{\mathit{n}}\right)}$  converges or diverges.

From this pattern, we can see that the $\mathit{k}$th$$ partial sum is given by the explicit formula,

${\mathit{S}}_{\mathit{k}}=\mathit{k}+{\displaystyle \sum _{\mathit{n}=1}^{\mathit{k}}\frac{1}{\mathit{n}} }$

Since this is a divergent sequence, we conclude that the sequence of partial sums diverges. Therefore, the infinite series ${\displaystyle \underset{\mathit{n}=1}{\overset{\infty }{ \sum }}\mathit{ }\left(\frac{\mathit{n}+5}{\mathit{n}}\right)}$ diverges.

Convergence of Geometric Series

A geometric series is a series of the form 

$\sum _{\mathit{n}=1}^{\infty }{\mathit{ar}}^{\mathit{n}-1}=\mathit{a}+\mathit{ar}+{\mathit{ar}}^2+{\mathit{ar}}^3+...$

If $\left|\mathit{r}\right|<1$, the series converges, and 

$\sum _{\mathit{n}=1}^{\infty }{\mathit{ar}}^{\mathit{n}-1}=\frac{\mathit{a}}{1-\mathit{r}}$ for $\left|\mathit{r}\right|<1$

If $\left|\mathit{r}\right|\ge 1$, the series diverges.

Example 2:

Check whether $\sum _{\mathit{n}=1}^{\infty }{\mathit{e}}^{2\mathit{n}}$ geometric series converges or diverges.

Solution: Writing this series as ${\mathit{e}}^2\sum _{\mathit{n}=1}^{\infty }{\left({\mathit{e}}^2\right)}^{\mathit{n}-1}$.

we can see that this is a geometric series where $\mathit{r}={\mathit{e}}^2>1$. Therefore, the series diverges.