Essential Concepts

• Given the infinite series
  
  $\displaystyle\sum _{n=1}^{\infty }{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\cdots$

  
  
  and the corresponding sequence of partial sums $\left\{{S}_{k}\right\}$ where
  

  ${S}_{k}=\displaystyle\sum _{n=1}^{k}{a}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\cdots +{a}_{k}$,

  
  
  the series converges if and only if the sequence $\left\{{S}_{k}\right\}$ converges.

• The geometric series $\displaystyle\sum _{n=1}^{\infty }a{r}^{n - 1}$ converges if $|r|<1$ and diverges if $|r|\ge 1$. For $|r|<1$,
  
  $\displaystyle\sum _{n=1}^{\infty }a{r}^{n - 1}=\frac{a}{1-r}$.
• The harmonic series
  
  $\displaystyle\sum _{n=1}^{\infty }\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots$

  
  
  diverges.

• A series of the form $\displaystyle\sum _{n=1}^{\infty }\left[{b}_{n}-{b}_{n+1}\right]=\left[{b}_{1}-{b}_{2}\right]+\left[{b}_{2}-{b}_{3}\right]+\left[{b}_{3}-{b}_{4}\right]+\cdots +\left[{b}_{n}-{b}_{n+1}\right]+\cdots$
  
  is a telescoping series. The $k\text{th}$ partial sum of this series is given by ${S}_{k}={b}_{1}-{b}_{k+1}$. The series will converge if and only if $\underset{k\to \infty }{\text{lim}}{b}_{k+1}$ exists. In that case,
  
  $\displaystyle\sum _{n=1}^{\infty }\left[{b}_{n}-{b}_{n+1}\right]={b}_{1}-\underset{k\to \infty }{\text{lim}}\left({b}_{k+1}\right)$.