Q. 116 - Precalculus Enhanced with Graphing Utilities

Question

Use a calculator to compute the values of 

$2+\frac{1}{2!}+\frac{1}{3!}+...+\frac{1}{n!}$

for $n=4,6,8,10$

Compare each result with e.

Step-by-Step Solution

Answer

When $n=4$ in the series, we get $2.708$.

When $n=6$ in the series, we get $2.717$.

When $n=8$ in the series, we get $2.71827$.

When $n=10$ in the series, we get $2.718281$.

On comparing the results with e,

Step 1. Given information.

Consider the given question,

$2+\frac{1}{2!}+\frac{1}{3!}+...+\frac{1}{n!}$

Let us say $S=2+\frac{1}{2!}+\frac{1}{3!}+...+\frac{1}{n!}$

Substitute $n=4$ in the series,

$$\begin{gathered} S=2+\frac{1}{2!}+\frac{1}{3!}+...+\frac{1}{4!} \\ =2+0.5+0.1667+0.0416 \\ =2.708 \end{gathered}$$

Substitute $n=6$ in the series,

$$\begin{gathered} S=2+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!} \\ =2+0.5+0.1667+0.0416+0.0083+0.0013 \\ =2.717 \end{gathered}$$

Step 2. Find the value of the series when n = 8 , 10 .

Substitute $n=8$ in the series,

$$\begin{gathered} S=2+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}+\frac{1}{7!}+\frac{1}{8!} \\ =2+0.5+0.1667+0.0416+0.0083+0.0013+0.000198+0.0000248 \\ =2.71827 \end{gathered}$$

Substitute $n=10$ in the series,

$$\begin{gathered} S=2+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}+\frac{1}{7!}+\frac{1}{8!}+\frac{1}{9!}+\frac{1}{10!} \\ =2+0.5+0.1667+0.0416+0.0083+0.0013+0.000198+0.0000248+0.000002755+0.00000027 \\ =2.718281 \end{gathered}$$

Step 3. Compare the results of the series.

On comparing the results with e is given below,