Chapter 11.4, Problem 14WE

To determine

To calculate:

The following series with sigma notation:

  $\frac{1}{5^3}+\frac{1}{5^4}+\frac{1}{5^5}+\mathrm{...}+\frac{1}{5^{15}}$

Answer to Problem 14WE

The sigma notation for $\frac{1}{5^3}+\frac{1}{5^4}+\frac{1}{5^5}+\mathrm{...}+\frac{1}{5^{15}}$ is ${\displaystyle \sum _{n=3}^{15}\frac{1}{5^n}}$ .

Explanation of Solution

Given information:

  $\frac{1}{5^3}+\frac{1}{5^4}+\frac{1}{5^5}+\mathrm{...}+\frac{1}{5^{15}}$

Calculation:

In the given series:

The first term is $3$ and the last term is $15$ . Here, the summand is $\frac{1}{5^n}$ .

Now, the upper and lower limits are calculated as:

Use the first term to calculate the lower limit:

  $\begin{array}{l}\frac{1}{5^n}=\frac{1}{5^3}\\ n=3\end{array}$

Use the last term to calculate the upper limit:

  $\begin{array}{l}\frac{1}{5^n}=\frac{1}{5^{15}}\\ n=15\end{array}$

Now, the sigma notation for the given series is expressed as:

  $\frac{1}{5^3}+\frac{1}{5^4}+\frac{1}{5^5}+\mathrm{...}+\frac{1}{5^{15}}={\displaystyle \sum _{n=3}^{15}\frac{1}{5^n}}$