Question

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

Short answer

The series is $\frac{1+x}{1-x}=1+\sum _{n=1}^{\infty }2x^n$.

The sum is given below.

$S_1=1+2x$

$S_2=1+2x+2x^2$

$S_3=1+2x+2x^2+2x^3$

The graphs are shown below.

Step-by-step solution

 Step 1: Given Information

The Maclaurin series.

Definition of the MaclaurinSeries.

A Maclaurin series is a function with an expansion series that gives the sum of the function's derivatives.

Find the series and sum.

The Maclaurin series is given below.

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

The general series is given below.

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

The sum is given below.

$S_1=1+2x$

$S_2=1+2x+2x^2$

$S_3=1+2x+2x^2+2x^3$

Plot the graphs.

The graph of the function with $S_1,S_2,S_3$is given below.