Chapter 10.2, Problem 26E

(a)

To determine

The first four terms and the general term for the Maclaurin series

Answer to Problem 26E

The value is $f\left(t\right)=2+2t^2+2t^4+\mathrm{......}+2t^{2n}+\mathrm{....}$

Explanation of Solution

Given information:

  $f\left(t\right)=\frac{2}{1-t^2}\text{and }{\displaystyle \underset{0}{\overset{x}{\int }}f\left(t\right)}dt$

Formula used:

  $\frac{1}{1-x}=1+x+x^2+x^3+\mathrm{....}+x^n+\mathrm{......}$

Calculation:

Maclaurin series of a function $\frac{1}{1-x}$ is given by

  $\frac{1}{1-x}=1+x+x^2+x^3+\mathrm{....}+x^n+\mathrm{......}$

Multiplying the whole series by 2 and substitute, $t^2$ for x in the Maclaurin series for $\frac{1}{1-x}$

  $\begin{array}{l}f\left(t\right)=\frac{2}{1-t^2}\\ =2\left[1+t^2+{\left(t^2\right)}^2+{\left(t^2\right)}^3+\mathrm{.....}+{\left(t^2\right)}^n+\mathrm{....}\right]\end{array}$

  $f\left(t\right)=2+2t^2+2t^4+\mathrm{......}+2t^{2n}+\mathrm{....}$

Conclusion:

The value is $f\left(t\right)=2+2t^2+2t^4+\mathrm{......}+2t^{2n}+\mathrm{....}$

(b)

To determine

The first four non zero terms.

Answer to Problem 26E

The value is $G\left(x\right)=2x+\frac{2x^3}{3}+\frac{2x^5}{5}+\mathrm{......}+\frac{2x^{2n+1}}{2n+1}+\mathrm{....}$

Explanation of Solution

Given information:

  $f\left(t\right)=\frac{2}{1-t^2}\text{and }{\displaystyle \underset{0}{\overset{x}{\int }}f\left(t\right)}dt$

Formula used:

  $G\left(x\right)={\displaystyle \underset{0}{\overset{x}{\int }}f\left(t\right)}dt$

Calculation:

Given,

  $G\left(x\right)={\displaystyle \underset{0}{\overset{x}{\int }}f\left(t\right)}dt$

Since,

  $G\left(0\right)=0$

The constant term is zero and we may find $G\left(x\right)$ by integrating the terms of the series for $f\left(t\right).G\left(x\right)={\displaystyle \underset{0}{\overset{x}{\int }}\left(2+2t^2+2t^4+\mathrm{.....}+2t^{2n}+\mathrm{.....}\right)dx}$

  $G\left(x\right)=2x+\frac{2x^3}{3}+\frac{2x^5}{5}+\mathrm{......}+\frac{2x^{2n+1}}{2n+1}+\mathrm{....}$

Conclusion:

The value is $G\left(x\right)=2x+\frac{2x^3}{3}+\frac{2x^5}{5}+\mathrm{......}+\frac{2x^{2n+1}}{2n+1}+\mathrm{....}$