5.2 Infinite series  (Page 7/16)

Key concepts

• Given the infinite series
  
  ${\displaystyle \sum }_{n=1}^{\infty }{a}_n=a_1+a_2+a_3+\text{⋯}$
  
  and the corresponding sequence of partial sums $\left\{S_k\right\}$ where
  
  $S_k={\displaystyle \sum }_{n=1}^k{a}_n=a_1+a_2+a_3+\text{⋯}+a_k,$
  
  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 $\left|r\right|<1$ and diverges if $\left|r\right|\ge 1.$ For $\left|r\right|<1,$
  
  ${\displaystyle \sum }_{n=1}^{\infty }a{r}^{n-1}=\frac{a}{1-r}.$
• The harmonic series
  
  ${\displaystyle \sum }_{n=1}^{\infty }\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\text{⋯}$
  
  diverges.
• A series of the form ${\displaystyle \sum _{n=1}^{\infty }[b_n-b_{n+1}]}=[b_1-b_2]+[b_2-b_3]+[b_3-b_4]+\text{⋯}+[b_n-b_{n+1}]+\text{⋯}$
  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,
  
  ${\displaystyle \sum _{n=1}^{\infty }[b_n-b_{n+1}]}=b_1-\underset{k\to \infty }{\text{lim}}\left(b_{k+1}\right).$

Key equations

• Harmonic series
  ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n}}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\text{⋯}$
• Sum of a geometric series
  ${\displaystyle \sum _{n=1}^{\infty }a{r}^{n-1}}=\frac{a}{1-r}\text{for}\left|r\right|<1$

Using sigma notation, write the following expressions as infinite series.

Compute the first four partial sums $S_1\text{,…},S_4$ for the series having $n\text{th}$ term $a_n$ starting with $n=1$ as follows.

In the following exercises, compute the general term $a_n$ of the series with the given partial sum $S_n.$ If the sequence of partial sums converges, find its limit $S.$

For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges.

Suppose that ${\displaystyle \sum _{n=1}^{\infty }{a}_n=1},$ that ${\displaystyle \sum _{n=1}^{\infty }{b}_n=\mathrm{-1}},$ that $a_1=2,$ and $b_1=\mathrm{-3}.$ Find the sum of the indicated series.

State whether the given series converges and explain why.

For $a_n$ as follows, write the sum as a geometric series of the form ${\displaystyle \sum _{n=1}^{\infty }a{r}^n}.$ State whether the series converges and if it does, find the value of ${\displaystyle \sum a_n}.$

Use the identity $\frac{1}{1-y}={\displaystyle \sum _{n=0}^{\infty }{y}^n}$ to express the function as a geometric series in the indicated term.

Evaluate the following telescoping series or state whether the series diverges.