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# Week 5 Thursday Problems
## Reading.
Please read sections 11.6 to 11.7. Do try the exercises for additional practice.
We focus now on ratio test and root test. Do as many additional mixed practices as you can.
Next week we will start 11.8 Power series.
## Problem.
1. Use ratio test (and possibly with some other techniques) to determine whether the following series is convergent or divergent.
1. $\displaystyle\sum_{n=1}^{\infty} \frac{n}{5^{n}}$
2. $\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1} \frac{3^{n}}{2^{n}n^{3}}$
3. $\displaystyle\sum_{n=0}^{\infty} \frac{1}{n!}$
4. $\displaystyle\sum_{n=1}^{\infty} \frac{10^{n}}{(n+1)4^{2n+1}}$
5. $\displaystyle\sum_{n=1}^{\infty} \frac{n\pi^{n}}{(-3)^{n+1}}$
6. $\displaystyle\sum_{n=1}^{\infty} \frac{\cos(n\pi /3)}{n!}$
7. $\displaystyle\sum_{n=1}^{\infty} \frac{n^{100}100^{n}}{n!}$
8. $\displaystyle1- \frac{2!}{1\cdot 3}+ \frac{3!}{1\cdot3\cdot5}-\frac{4!}{1\cdot3\cdot5\cdot 7}+\cdots+(-1)^{n-1} \frac{n!}{1\cdot 3\cdot 5\cdot\cdots\cdot(2n-1)}+\cdots$
9. $\displaystyle \frac{2}{3}+ \frac{2 \cdot 5}{3\cdot 5}+ \frac{2\cdot 5\cdot 8}{3\cdot 5\cdot 7}+ \frac{2\cdot 5\cdot 8\cdot 11}{3\cdot 5\cdot 7\cdot 9}+\cdots$
10. $\displaystyle\sum_{n=1}^{\infty} \frac{2\cdot4\cdot6\cdot\cdots\cdot(2n)}{n!}$
11. $\displaystyle\sum_{n=1}^{\infty}(-1)^{n} \frac{2^{n}n!}{5\cdot8\cdot11\cdot\cdots\cdot(3n+2)}$
2. Use root test to determine whether the series converges or diverges.
1. $\displaystyle\sum_{n=1}^{\infty}\left( \frac{n^{2}+1}{2n^{2}+1} \right)^{n}$
2. $\displaystyle\sum_{n=1}^{\infty}\frac{(-2)^{n}}{n^{n}}$
3. $\displaystyle\sum_{n=2}^{\infty} \frac{(-1)^{n+1}}{(\ln(n))^{n}}$
4. $\displaystyle\sum \left( \frac{-2n}{n+1} \right)^{5n}$
5. $\displaystyle\sum_{n=1}^{\infty}\left( 1+ \frac{1}{n} \right)^{(n^{2})}$
3. Determine whether the following series is absolutely convergent, conditionally convergent, or divergent.
1. $\displaystyle\sum_{n=2}^{\infty} \frac{(-1)^{n}\ln(n)}{n}$
2. $\displaystyle\sum_{n=1}^{\infty} \left( \frac{1-n}{2+3n} \right)^{n}$
3. $\displaystyle\sum_{n=1}^{\infty} \frac{(-9)^{n}}{n 10^{n+1}}$
4. $\displaystyle\sum_{n=2}^{\infty} \left( \frac{n}{\ln n} \right)^{n}$
5. $\displaystyle\sum_{n=1}\frac{(-1)^{n}\arctan(n)}{n^{2}}$
4. Do as many practice problem from your textbook section 11.7 as you can.
5. Make a cheat sheet that organizes all your tests and strategies.
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