# Section 6.3 Calculus 2

Manipulating Power Series

## 6.3 Manipulating Power Series

### 6.3.1 Multiplication and Composition with Power Series

• Let $$p(x)$$ be a polynomial. If $$\sum_{n=0}^\infty c_n(x-a)^n$$ is a power series converging to $$f(x)$$, then $$\sum_{n=0}^\infty p(x)c_n(x-a)^n$$ is a power series converging to $$p(x)f(x)$$ with the same domain of convergence.
• Example. Find the Maclaurin series for $$x^3\cos(x)$$.
• Let $$p(x)$$ be a polynomial. If $$\sum_{n=0}^\infty c_n(x-a)^n$$ is a power series converging to $$f(x)$$, then $$\sum_{n=0}^\infty c_n(p(x)-a)^n$$ is a power series converging to $$f(p(x))$$ with an appropriate domain of convergence.
• Example. Find the Maclaurin series for $$\exp(x^2)$$.
• Example. Find the Maclaurin series for $$x\sin(2x)$$.

### 6.3.2 Differentiating and Integrating Power Series

• If $$\sum_{n=0}^\infty c_n(x-a)^n$$ is a power series converging to $$f(x)$$ on an open interval, then $$\sum_{n=0}^\infty c_n n(x-a)^{n-1}$$ is a power series converging to $$f’(x)$$ with the same domain of convergence.
• Example. Prove that $$\frac{d}{dx}[e^x]=e^x$$.
• Example. Find the Maclaurin series for $$\frac{1}{(1-x)^2}$$ where $$|x|<1$$.
• If $$\sum_{n=0}^\infty c_n(x-a)^n$$ is a power series converging to $$f(x)$$ on an open interval, then $$\sum_{n=0}^\infty c_n \frac{(x-a)^{n+1}}{n+1} + C$$ is a power series converging to $$\int f(x)\,dx$$ with the same domain of convergence.
• Example. Find a power series converging to $$\ln(x)$$ where $$0<x<2$$.

### Review Exercises

1. Find a power series converging to $$\frac{x^2}{1-x}$$ for $$-1<x<1$$.
2. Find a power series converging to $$x^3\cos(x)$$ for all $$x$$.
3. Find the Taylor series generated by $$(x-\pi/2)\sin(x)$$ at $$\pi/2$$.
4. Find the Maclaurin series generated by $$\sin(x^3)$$.
5. Find a power series converging to $$\frac{2}{2-x}$$ for $$-2<x<2$$. (Hint: start with $$\frac{1}{1-x}=\frac{2}{2-2x}$$.)
6. Find the Maclaurin series generated by $$3x^2\cos(x^3)$$.
7. Find a power series converging to $$\tan^\leftarrow(x)$$ for $$-1<x<1$$. (Hint: use $$\frac{1}{1-(-x^2)}$$.)

Solutions

### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 9.8, 9.9