Q. 23 - Calculus

Question

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

$\sum _{j=0}^{\infty }{\left(-1\right)}^{j+1}$

Step-by-Step Solution

Answer

Ans: The five terms of the series are $-1,1,-1,1,-1$

Step 1. Given information:

$\sum _{j=0}^{\infty }{\left(-1\right)}^{j+1}$

Step 2. Finding the first term of the series:

The first term of the series $\sum _{j=0}^{\infty }(-1{)}^{j+1}$ is obtained by substituting $j=0$ in $(-1{)}^{j+1}$. Therefore, the value at $j=0$ is:

$(-1{)}^{0+1}=(-1{)}^1$ (Substituting)

$=-1$

The first term of the series $\sum _{j=0}^{\infty }(-1{)}^{j+1}$ is -1.

Step 3. Finding the second term of the series:

The second term of the series $\sum _{j=0}^{\infty }(-1{)}^{j+1}$ is obtained by substituting $j=1$ in $(-1{)}^{j+1}$. Therefore, the value at $j=1$ is:

$(-1{)}^{1+1}=(-1{)}^2$(Substituting)

=1

The second term of the series $\sum _{j=0}^{\infty }(-1{)}^{j+1}$ is 1 .

Step 4. Finding the third term of the series:

The third term of the series $\sum _{j=0}^{\infty }(-1{)}^{j+1}$ is obtained by substituting $j=2$ in $(-1{)}^{j+1}$. Therefore, the value at $j=2$ is:

$(-1{)}^{2+1}=(-1{)}^3$ (Substituting)

=-1

The third term of the series $\sum _{j=0}^{\infty }(-1{)}^{j+1}$ is -1.

Step 5. Finding the fourth term of the series:

The fourth term of the series $\sum _{j=0}^{\infty }(-1{)}^{j+1}$ is obtained by substituting $j=3$ in $(-1{)}^{j+1}$. Therefore, the value at $j=3$ is:

$(-1{)}^{3+1}=(-1{)}^4$( Substituting )

=1

The fourth term of the series $\sum _{j=0}^{\infty }(-1{)}^{j+1}$ is 1 .

Step 6. Finding the fifth term of the series:

The fifth term of the series $\sum _{j=0}^{\infty }(-1{)}^{j+1}$ is obtained by substituting $j=4$ in $(-1{)}^{j+1}$. Therefore, the value at $j=4$ is:

$(-1{)}^{4+1}=(-1{)}^5$ (Substituting)

=-1

The fifth term of the series $\sum _{j=0}^{\infty }(-1{)}^{j+1}$ is -1.