Question

In Problems 51–60, write out each sum.

$\sum _{k=0}^n\frac{1}{3^k}$

Short answer

The required sum is$\sum _{k=0}^n\frac{1}{3^k}=\left\{1+\frac{1}{3}+\frac{1}{9}+......+\frac{1}{3^n}\right\}$.

Step-by-step solution

Step 1. Write the given information.

The given summation notation is:

$\sum _{k=0}^n\frac{1}{3^k}$

Step 2. Write out the sum.

To get the sum, we need to substitute $0,1,2,3,......,n$for k,in the given sequence $\sum _{k=0}^n\frac{1}{3^k}$,

$$\begin{gathered} \sum _{k=0}^n\frac{1}{3^k}=\left\{\frac{1}{3^0}+\frac{1}{3^1}+\frac{1}{3^2}+......+\frac{1}{3^n}\right\} \\ =\left\{1+\frac{1}{3}+\frac{1}{9}+......+\frac{1}{3^n}\right\} \end{gathered}$$

Therefore,$\sum _{k=0}^n\frac{1}{3^k}=\left\{1+\frac{1}{3}+\frac{1}{9}+......+\frac{1}{3^n}\right\}$.