Chapter 11.5, Problem 20E

In Exercises $5-20$, find the Taylor series at $x=0$ of the given function. Use suitable operation (differentiation, substitution, etc.) on the Taylor series at $x=0$ of $\frac{1}{1-x},e^x,\text{or}\cos x$. These series are derived in Example 1 and 2 and check your understanding problem 2.

$x\sin x^2$

Example 1:

The Taylor series expansion of $\frac{1}{1-x}$ at $x=0$ is $1+x+x^2+x^3+x^4+\cdot \cdot \cdot$

Example 2:

The Taylor series expansion of $e^x$ at $x=0$ is $1+x+\frac{1}{2!}{x}^2+\frac{1}{3!}{x}^3+\frac{1}{4!}{x}^4+\cdot \cdot \cdot$

Check your understanding problem 2:

The Taylor series expansion of $\cos x$ at $x=0$ is $1-\frac{1}{2!}{x}^2+\frac{1}{4!}{x}^4-\cdot \cdot \cdot$