# Section 6.1 Calculus 2

Power Series

## 6.1 Power Series

### 6.1.1 Definition

• A power series is a function defined by $$f(x)=\sum_{n=0}^\infty c_n(x-a)^n= c_0+c_1(x-a)+c_2(x-a)^2+c_3(x-a)^3+\dots$$ for a coefficient sequence $$\<c_n\>_{n=0}^\infty$$ and center $$a$$.
• Example Expand the first four terms of the power series $$\sum_{m=0}^\infty (2m+1)x^m$$.
• Example Expand the first four terms of the power series $$\sum_{k=2}^\infty \frac{x^{2k}}{k!}$$.
• The geometric series formula $$\sum_{n=0}^\infty ar^n$$ allows us to simplify certain series where $$|r|<1$$.
• Example Simplify $$f(x)=\sum_{n=0}^\infty x^n=1+x+x^2+x^3+\dots$$ with domain $$|x|<1$$.
• Example Simplify $$p(x)=\sum_{j=1}^\infty (\frac{x}{3})^{2j}= \frac{x^2}{9}+\frac{x^4}{81}+\frac{x^6}{729}+\dots$$ with domain $$|x|<3$$.

### 6.1.2 Domains of Power Series

• The domain of a power series may be determined by applying the Root or Ratio Test to determine for which $$x$$ values the series converges. On the endpoints where these tests are inconclusive, other techniques must be used.
• Example Find the domain of $$f(x)=\sum_{n=1}^\infty\frac{x^n}{n}=x+\frac{x^2}{2}+\frac{x^3}{3}+\dots$$.
• Example Find the domain of $$h(x)=\sum_{n=0}^\infty\frac{(3-2x)^n}{n!}= 1+(3-2x)+\frac{(3-2x)^2}{2}+\frac{(3-2x)^3}{6}+\dots$$.
• Example Find the domain of $$g(x)=\sum_{k=2}^\infty\frac{(3x)^n}{n\ln n}= \frac{9x^2}{2\ln2}+\frac{27x^3}{3\ln3}+\frac{81x^4}{4\ln4}+\dots$$.

### Review Exercises

1. Expand the first four terms of the power series $$\sum_{m=0}^\infty 3^{m+1}x^m$$.
2. Expand the first four terms of the power series $$\sum_{k=1}^\infty \frac{(-x)^k}{k+1}$$.
3. Expand the first four terms of the power series $$\sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}$$.
4. Simplify $$q(x)=\sum_{n=1}^\infty (1-x)^n=(1-x)+(1-x)^2+(1-x)^3+\dots$$ with domain $$|1-x|<1$$.
5. Simplify $$g(x)=\sum_{j=0}^\infty (2x)^{2j+1}= 2x+8x^3+32x^5+128x^7+\dots$$ with domain $$|x|<\frac{1}{2}$$.
6. Find the domain of $$z(x)=\sum_{n=2}^\infty(-1)^n\frac{x^n}{2n}= \frac{x^2}{4}-\frac{x^3}{6}+\frac{x^4}{8}-\frac{x^5}{10}+\dots$$.
7. Find the domain of $$f(x)=\sum_{i=0}^\infty\frac{(3x)^i}{(2i)!}= 1+\frac{3}{2}x+\frac{9}{24}x^2+\dots$$.
8. Find the domain of $$h(x)=\sum_{k=0}^\infty\frac{(x-2)^k}{k^2+1}= 1+\frac{x-2}{2}+\frac{(x-2)^2}{5}+\frac{(x-2)^3}{10}+\dots$$.
9. Find the domain of $$g(x)=\sum_{m=3}^\infty(\frac{1}{m}-\frac{1}{m+1})x^m$$.
10. Expand the first four terms of the power series $$\sum_{k=0}^\infty \frac{(-x)^{2k+1}}{(2k)!}$$.
11. Simplify $$f(x)=\sum_{n=1}^\infty (-x)^{n-1}=1-x+x^2-x^3+\dots$$ with domain $$|x|<1$$.
12. Find the domain of $$f(x)=\sum_{m=2}^\infty\frac{(-2x)^m}{m}= \frac{4x^2}{2}-\frac{8x^3}{3}+\frac{16x^4}{4}-\frac{32x^5}{5}+\dots$$.

Solutions

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 9.7