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# Week 6 Monday Problems
## Reading.
Please read sections 11.8 to 11.10 for this week. These are about power series and Taylor and Maclaurin series.
Don't forget next Monday we have **EXAM 2 in class**. You get to use one page of cheat sheet (both sides).
## Problems.
1. Describe the following:
1. What is a power series?
2. Given a power series, what is its interval of convergence?
3. Given a power series, what is its radius of convergence?
4. How might one find the interval of convergence and radius of convergence for a power series?
2. For each of the power series below, find both (1) the radius of convergence and (2) the interval of convergence.
1. $\displaystyle\sum_{n=1}^{\infty} \frac{x^{n}}{n}$
2. $\displaystyle\sum_{n=1}^{\infty}\sqrt{n}x^{n}$
3. $\displaystyle\sum_{n=1}^{\infty} \frac{n}{5^{n}}x^{n}$
4. $\displaystyle\sum_{n=1}^{\infty} \frac{x^{n}}{n3^{n}}$
5. $\displaystyle\sum_{n=1}^{\infty} \frac{x^{n}}{2n-1}$
6. $\displaystyle\sum_{n=0}^{\infty} \frac{x^{n}}{n!}$
7. $\displaystyle\sum_{n=0}^{\infty} \frac{x^{n}}{(2n)!}$
8. $\displaystyle\sum_{n=1}^{\infty} \frac{x^{n}}{n^{4}4^{n}}$
9. $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n}4^{n}}{\sqrt{n}}x^{n}$
10. $\displaystyle\sum_{n=1}^{\infty} \frac{n}{2^{n}(n^{2}+1)}x^{n}$
11. $\displaystyle\sum_{n=1}^{\infty} \frac{(x-2)^{n}}{n^{2}+1}$
12. $\displaystyle\sum_{n=3}^{\infty} \frac{\ln(n)}{n}x^{n}$
13. $\displaystyle\sum_{n=1}^{\infty} \frac{(5x-4)^{n}}{n^{3}}$
14. $\displaystyle\sum_{n=1}^{\infty} \frac{x^{n}}{1\cdot3\cdot5\cdot\cdots\cdot(2n-1)}$
3. If $\displaystyle\sum_{n=0}^{\infty}c_{n}4^{n}$ is convergent, for some coefficients $c_{n}$, can we say anything about the convergence of the following series? Hint: Come up with a power series and say something about that power series.
1. $\displaystyle\sum_{n=0}^{\infty}c_{n}(-2)^{n}$
2. $\displaystyle\sum_{n=0}^{\infty}c_{n}(-4)^{n}$
4. Suppose $\displaystyle\sum_{n=0}^{\infty}c_{n}x^{n}$ converges at $x=-4$ and diverges at $x=6$. What can we say about the convergence or divergence of the following series?
1. $\displaystyle\sum_{n=0}^{\infty}c_{n}$
2. $\displaystyle\sum_{n=0}^{\infty}c_{n}8^{n}$
3. $\displaystyle\sum_{n=0}^{\infty}c_{n}(-3)^{n}$
4. $\displaystyle\sum_{n=0}^{\infty}(-1)^{n}c_{n}9^{n}$
5. Give an example of a power series whose interval of convergence is of the following interval form, where $p < q$ two real numbers:
1. $(p,q)$
2. $(p,q]$
3. $[p,q)$
4. $[p,q]$
6. Is there a power series whose interval of convergence is $[0,\infty)$?
7. Give a concrete example of a power series whose interval of convergence is $(3,7]$. Hint: Use some of the known power series that you have and transform them to it.
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