Chapter 8.1, Problem 133E

To determine

To write: the first five terms of the sequence.

Answer to Problem 133E

The first five terms are

  $x,\frac{x^2}{2},\frac{x^3}{6},\frac{x^4}{24},\frac{x^5}{120}$

Explanation of Solution

Given information:

Given sequence

  $a_n=\frac{x^n}{n!}$

Concept used:

Factorial of a number n can be simplified as

  $n!=n\left(n-1\right)\left(n-2\right)\left(n-3\right)\mathrm{...3}\cdot 2\cdot 1$

Where $n$ is a positive integer.

Calculation:

Let, $a_i$ is the $i^{th}$ term of the sequence then,

The first five terms given by the sequence $a_n=\frac{x^n}{n!}$ are,

  $\begin{array}{l}{a}_1=\frac{x^1}{1!}=x\\ a_2=\frac{x^2}{2!}=\frac{x^2}{2\times 1}=\frac{x^2}{2}\\ a_3=\frac{x^3}{3!}=\frac{x^3}{3\times 2\times 1}=\frac{x^3}{6}\\ a_4=\frac{x^4}{4!}=\frac{x^4}{4\times 3\times 2\times 1}=\frac{x^4}{24}\\ a_5=\frac{x^5}{5!}=\frac{x^5}{5\times 4\times 3\times 2\times 1}=\frac{x^5}{120}\end{array}$

Hence, the first five terms are

  $x,\frac{x^2}{2},\frac{x^3}{6},\frac{x^4}{24},\frac{x^5}{120}$