Q. 22 - Calculus

Question

In Exercises 21–28 provide the first five terms of the series.

$\sum _{n=1}^{\infty } \frac{n+1}{n!}$

Step-by-Step Solution

Answer

Ans: The five terms of the series are $2,\frac{3}{2},\frac{2}{3},\frac{5}{24},\frac{1}{20}$

Step 1. Given information:

$\sum _{n=1}^{\infty } \frac{n+1}{n!}$

Step 2. Finding the first term of the series:

The first term of the series $\sum _{n=1}^{\infty }\frac{n+1}{n!}$ is obtained by substituting $n=1$ in $\frac{n+1}{n!}$. Therefore, the value at $n=1$ is:

$\frac{n+1}{n!}=\frac{1+1}{1!}$ (Substituting)

$=2$

The first term of the series is 2 .

Step 3. Finding the second term of the series:

The second term of the series $\sum _{n=1}^{\infty }\frac{n+1}{n!}$ is obtained by substituting $n=2$ in $\sum _{n=1}^{\infty }\frac{n+1}{n!}$. Therefore, the value at $n=2$ is:

$\frac{n+1}{n!}=\frac{2+1}{2!}$ (Substituting)

$=\frac{3}{2}$

The second term of the series $\sum _{n=1}^{\infty }\frac{n+1}{n!}$ is $\frac{3}{2}$.

Step 4. Finding the third term of the series:

The third term of the series $\sum _{n=1}^{\infty }\frac{n+1}{n!}$ is obtained by substituting $n=3$ in $\frac{n+1}{n!}$. Therefore, the value at $n=3$is:

$\frac{3+1}{3!}=\frac{4}{3\times 2\times 1}$ (Substituting)

$=\frac{2}{3}$

The third term of the series $\sum _{n=1}^{\infty }\frac{n+1}{n!}$ is $\frac{2}{3}$.

Step 5. Finding the fourth term of the series:

The fourth term of the series $\sum _{n=1}^{\infty }\frac{n+1}{n!}$ is obtained by substituting $n=4$ in $\frac{n+1}{n!}$ Therefore, the value at $n=4$ is:

$\frac{4+1}{4!}=\frac{5}{4\times 3\times 2\times 1}$( Substituting)

$=\frac{5}{24}$

The fourth term of the series $\sum _{n=1}^{\infty }\frac{n+1}{n!}$ is $\frac{5}{24}$.

Step 6. Finding the fifth term of the series:

The fifth term of the series$\sum _{n=1}^{\infty }\frac{n+1}{n!}$ is obtained by substituting $n=5$ in$\frac{n+1}{n!}$. Therefore, the value at $n=5$ is:

$\frac{5+1}{5!}=\frac{6}{5\times 4\times 3\times 2\times 1}$ (Substituting)

$=\frac{1}{20}$

The fifth term of the series $\sum _{n=1}^{\infty }\frac{n+1}{n!}$ is $\frac{1}{20}$.