Chapter F, Problem 7E

To determine

To write: The sum ${\displaystyle \sum _{i=1}^n{i}^{10}}$ is in expanded form.

Answer to Problem 7E

The expanded form of the sum ${\displaystyle \sum _{i=1}^n{i}^{10}}$ is $1^{10}+2^{10}+3^{10}+\cdots +n^{10}$.

Explanation of Solution

Definition used:

If $a_m,a_{m+1},\mathrm{...},a_n$ are real numbers and m and n are integers such that $m\le n$, then ${\displaystyle \sum _{i=m}^n{a}_i=a_m+}{a}_{m+1}+a_{m+2}+\cdots +a_{n-1}+a_n$.

Calculation:

By the above definition, the expression ${\displaystyle \sum _{i=1}^n{i}^{10}}$ simplified as follows.

${\displaystyle \sum _{i=1}^n{i}^{10}}=1^{10}+2^{10}+3^{10}+\cdots +n^{10}$

Thus, the expanded form of the sum ${\displaystyle \sum _{i=1}^n{i}^{10}}$ is $1^{10}+2^{10}+3^{10}+\cdots +n^{10}$.