Q. 27 - Calculus

Question

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

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

Step-by-Step Solution

Answer

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

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

Step 1: Given information

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

Step 2: Write the series in term of summation

Rewrite the given series in term of summation.

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

Step 3: Write the conclusion

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