Chapter 11.5, Problem 2E

To determine

To fill:The value of ${\displaystyle \sum _{i=1}^n{i}^3}$ .

Answer to Problem 2E

Thevalue of ${\displaystyle \sum _{i=1}^n{i}^3}$ is $\frac{n^2{\left(n+1\right)}^2}{4}$ .

Explanation of Solution

Given information:The summation notation is ${\displaystyle \sum _{i=1}^n{i}^3}$ .

The symbol of summation is used to represent the terms of a series in a particular manner.

The value of ${\displaystyle \sum _{i=1}^n{i}^3}$ is:

  $\begin{array}{c}{\displaystyle \sum _{i=1}^n{i}^3}=1+2^3+3^3+\cdots +n^3\\ =\frac{n^2{\left(n+1\right)}^2}{4}\end{array}$

Therefore, the value of ${\displaystyle \sum _{i=1}^n{i}^3}$ is $\frac{n^2{\left(n+1\right)}^2}{4}$ .