Question

Write each of the sums in Exercises 27 in sigma notation. Identify $m$, $n$, and $a_k$in each problem.

$\frac{1}{n}+\frac{2}{n}+\frac{3}{n}+\cdots +\frac{n}{n}$

Short answer

Here,$m=1,n=n$and$a_k=\frac{k}{n}$

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

Step-by-step solution

Given information

The given series is $\frac{1}{n}+\frac{2}{n}+\frac{3}{n}+\cdots +\frac{n}{n}$.

Write the series in term of summation

Rewrite the given series in term of summation.

$\begin{array}{rcl}\frac{1}{n}+\frac{2}{n}+\frac{3}{n}+\cdots +\frac{n}{n}& =& \frac{1+2+3+.....+n}{n}\\ & =& \frac{\sum _{k=1}^{n}k}{n}\\ & =& \sum _{k=1}^n\frac{k}{n}\end{array}$

Write the conclusion

The summation is$\sum _{k=1}^n\frac{k}{n}$.