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# Week 7 Tuesday Problems
## Reading.
Please read sections 11.8 to 11.10. These are about power series and Taylor and Maclaurin series.
Do Exam 2 corrections.
## Problems.
1. Give an example power series whose interval of convergence is **exactly** $(3,11]$.
2. Starting with the known power series representation of $\displaystyle \frac{1}{1-x}$ with $\displaystyle\sum_{n=0}^{\infty}x^{n}$ with interval of convergence $(-1,1)$, manipulate them to obtain a power series representation for the following functions **centered at 0**. Also indicate what is the interval of convergence. Hint: Remember the theorem "power series are nice"
1. $\displaystyle \frac{1}{1+x}$
2. $\displaystyle \frac{1}{1+x^{2}}$
3. $\displaystyle \frac{1}{1-x^{5}}$
4. $\displaystyle \frac{1}{(1-x)^{2}}$
5. $\displaystyle \ln(1+x)$
6. $\displaystyle \arctan(x)$
7. $\displaystyle \frac{5}{1-4x^{2}}$
8. $\displaystyle \frac{x-1}{x+2}$
9. $\displaystyle \frac{x^{2}}{x^{4}+16}$
3. Using partial fractions and manipulating geometric series, find a power series representation of $\displaystyle \frac{2x-4}{x^{2}-4x+3}$ **centered at 0**. What is the interval of convergence? Can you make it center at $2$?
4. By differentiating and manipulating our known geometric series:
1. Find a power series representation for $\displaystyle f(x)=\frac{1}{(1+x)^{2}}$. What is its radius of convergence?
2. Use previous result to find a power series representation for $\displaystyle f(x)=\frac{1}{(1+x)^{3}}$
3. Use previous result to find a power series representation for $\displaystyle f(x)=\frac{x^{2}}{(1+x)^{3}}$
5. By integrating and manipulating our known geometric series:
1. Find a power series representation for $f(x) = \ln(1-x)$, with **center 0**. What is its radius of convergence?
2. Use previous result to find a power series representation for $f(x) = x\ln(1-x)$.
3. Use the first part to express $\ln(2)$ as an infinite series. Hint: Try plugging in $\displaystyle x=\frac{1}{2}$, and think about properties of natural log.
6. Find a power series representation for the following **with center 0**, and find their radius of convergence. Then try to find a power series representation of them **with center 3**:
1. $\displaystyle \frac{x}{(1+4x)^{2}}$
2. $\displaystyle \left( \frac{x}{2-x} \right)^{3}$
3. $\displaystyle \frac{1+x}{(1-x)^{2}}$
4. $\displaystyle \ln(5-x)$
7. Evaluate the following improper integrals using power series, and what are their radius of convergence?
1. $\displaystyle \int \frac{x}{1-x^{8}}dx$
2. $\displaystyle \int x^{2}\ln(1+x)dx$
3. $\displaystyle \int \frac{\arctan(x)}{x}dx$
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