Question

$\mathit{l}\mathit{n}\frac{\mathbf{1}\mathbf{+}\mathbf{x}}{\mathbf{1}\mathbf{-}\mathbf{x}}$

Short answer

The Maclaurin series, i.e., $\ln \frac{1+x}{1-x}=2x+\frac{2x^3}{3}+\frac{2x^5}{5}+\dots$.

The general term, i.e., $\ln \frac{1+x}{1-x}=2\sum _{n=0}^{\infty }\frac{x^{2n+1}}{2n+1}$.

The Maclaurin series using MATLAB tool, i.e.,

Graph of the function and $s_1$ together, i.e.,

Graph of the function and $s_2$ together, i.e.,

Graph of the function and $s_3$together, i.e.,

Step-by-step solution

Given Information.

The given function.

$\ln \frac{1+x}{1-x}$

Definition of Maclaurin series.

All of the terms in a Maclaurin series are nonnegative integer powers of the variable. It's a Taylor series expansion of a function with a value of around 0.

Find the Maclaurin series for ln1+x1-x .

Use the Maclaurin series of $\ln (1+x)$ and replace$x$ by $-x$,

$f\left(x\right)=f\left(0\right)+\frac{f^{\text{'}}\left(0\right)}{1!}\left(x\right)+\frac{f^{\text{'}\text{'}}\left(0\right)}{2!}\left(x^2\right)+\frac{f^{\text{'}\text{'}\text{'}}\left(0\right)}{3!}\left(x^3\right)+\dots$

$\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\dots$

localid="1657466419308" $\ln (1-x)=-x-\frac{x^2}{2}-\frac{x^3}{3}-\frac{x^4}{4}+\dots$

Further, subtract $\ln (1-x)$from $\ln (1+x)$.

$\ln (1+x)-\ln (1-x)=\left[x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\dots \right]-\left[-x-\frac{x^2}{2}-\frac{x^3}{3}-\frac{x^4}{4}+\dots \right]$

$\ln \frac{1+x}{1-x}=2x+\frac{2x^3}{3}+\frac{2x^5}{5}+\dots$

Write the general term of ln1+x1-x  .

Use the general term of $\ln (1+x)$ and replace $x$by $-x$.

$\ln \frac{1+x}{1-x}=2\sum _{n=0}^{\infty }\frac{x^{2n+1}}{2n+1}$

Use MATLAB tool to find Maclaurin series.

Find first few terms of Maclaurin series.

Approximate partial sums of the series.

Write partial sums of the series $[s_1,s_2,s_3]$.

$$\begin{gathered} s_1=2x \\ {s}_2=2x+\frac{2x^3}{3} \\ {s}_3=2x+\frac{2x^3}{3}+\frac{2x^5}{5}+\dots \end{gathered}$$

Plot the function ln1+x1-x and s1together.

Graph of the function and $s_1$ together are shown as:

Plot the function ln1+x1-x  and s2  together.

Graph of the function and $s_2$together are shown as:

Plot the function  ln1+x1-x and s3 together.

Graph of the function and $s_3$together are shown as: