Chapter 9, Problem 62RE

$\text{(a)}$

To determine

To find: The Maclaurin series.

Answer to Problem 62RE

The answer:

Explanation of Solution

Given information:

The equation is $f(x)=x^2/(1+x)$ .

Calculation:

The Maclaurin series for $1/(1+x)$ is the following for each $\left|x\right|<1$ .

So,

  

By multiplying the series in the previous step by $x^2$ and obtain the Maclaurin series generated by the given function,

  

Therefore, the required Maclaurin series is

$\text{(b)}$

To determine

To find: The series converge at $x=1$ or not.

Answer to Problem 62RE

The answer: The series does not converge at $x=1$ .

Explanation of Solution

Given information:

The equation is $f(x)=x^2/(1+x)$ .

Calculation:

The series at $x=1$ ,

So,

  

which diverges, because the general term of the series does not approach zero as $n$ approaches infinity.

Therefore, the required series does not converge at $x=1$ .