Question

In Problems 61–70, express each sum using summation notation.

$\frac{1}{e}+\frac{2}{e^2}+\frac{3}{e^3}+.........+\frac{n}{e^n}$

Short answer

The required summation notation is $\frac{1}{e}+\frac{2}{e^2}+\frac{3}{e^3}+.........+\frac{n}{e^n}=\sum _{k=1}^n\frac{k}{e^k}$

Step-by-step solution

Step 1. Write the given information.

The given sum is:

$\frac{1}{e}+\frac{2}{e^2}+\frac{3}{e^3}+.........+\frac{n}{e^n}$

Step 2. Write the sum using summation notation.

As we can see that the sum $\frac{1}{e}+\frac{2}{e^2}+\frac{3}{e^3}+.........+\frac{n}{e^n}$has n terms, each term represents the form $\frac{k}{e^k}$, and starts at$k=1$ and ends at $k=n$.

Therefore,$\frac{1}{e}+\frac{2}{e^2}+\frac{3}{e^3}+.........+\frac{n}{e^n}=\sum _{k=1}^n\frac{k}{e^k}$.