Chapter F, Problem 19E

To determine

To write: The sum $x+x^2+x^3+\cdots +x^n$ in sigma notation.

Answer to Problem 19E

The sum $x+x^2+x^3+\cdots +x^n$ expressed in sigma notation as ${\displaystyle \sum _{i=1}^n{x}^i}$.

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:

Here, the sum $x+x^2+x^3+\cdots +x^n$ is the sum of $n$  numbers and number is of the form $x^i$, where $i$ varies from 0 to $n$.

Thus, by the above definition the sum $x+x^2+x^3+\cdots +x^n$ expressed as follows.

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

Thus, the sum $x+x^2+x^3+\cdots +x^n$ expressed in sigma notation as ${\displaystyle \sum _{i=1}^n{x}^i}$.