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# Week 5 Tuesday Problems
## Reading.
Please read sections 11.4 to 11.5. Do try the exercises for additional practice.
We focus now on comparison tests: Direct comparison test and limit comparison test. There is also an extended version of limit comparison test that deals with the limit cases of $0$ and $\infty$.
# Problems.
(These look like a lot but the goal is to practice enough to get comfortable with them!)
1. Determine whether the following series converge or diverge.
1. $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{3}+8}$
2. $\displaystyle\sum_{n=1}^{\infty} \frac{n+1}{n \sqrt{n}}$
3. $\displaystyle\sum_{n=1}^{\infty} \frac{9^{n}}{3+10^{n}}$
4. $\displaystyle\sum_{n=2}^{\infty} \frac{1}{\ln(n)}$
5. $\displaystyle\sum_{n=1}^{\infty} \frac{\sqrt[3]{k}}{\sqrt{k^{3}+4k+3}}$
6. $\displaystyle \sum_{n=1}^{\infty} \frac{1+\cos(n)}{e^{n}}$
7. $\displaystyle\sum_{n=1}^{\infty} \frac{4^{n+1}}{3^{n}-2}$
8. $\displaystyle\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{2}+1}}$
9. $\displaystyle\sum_{n=1}^{\infty} \frac{n+1}{n^{3}+n}$
10. $\displaystyle\sum_{n=1}^{\infty} \frac{\sqrt{1+n}}{2+n}$
11. $\displaystyle\sum_{n=1}^{\infty} \frac{5+2n}{(1+n^{2})^{2}}$
12. $\displaystyle\sum_{n=1}^{\infty} \frac{e^{n}+1}{ne^{n}+1}$
13. $\displaystyle\sum_{n=1}^{\infty} \frac{2+\sin(n)}{n^{2}}$
14. $\displaystyle\sum_{n=1}^{\infty}\left( 1+\frac{1}{n} \right)^{2}e^{-n}$
15. $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n!}$
16. $\displaystyle\sum_{n=1}^{\infty}\sin\left( \frac{1}{n} \right)$: Hint: Take an appropriate limit to compare...
17. $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n}\tan\left( \frac{1}{n} \right)$
2. We also have extended version of the limit comparison test (limit comparison test "pro"). Use them to determine whether the following series converge or diverge
1. $\displaystyle\sum_{n=1}^{\infty}\frac{\ln n}{n^{3}}$
2. $\displaystyle\sum_{n=1}^{\infty}\left( 1-\cos\left( \frac{1}{n^{2}} \right) \right)$
3. $\displaystyle\sum_{n=2}^{\infty} \frac{1}{\ln n}$
4. $\displaystyle\sum_{n=1}^{\infty} \frac{\ln(n)}{n}$
3. Given an example of series $\sum a_{n}$ and $\sum b_{n}$ with positive terms ($a_{n}\ge0,b_{n}\ge0$, where $\displaystyle\lim_{n\to\infty} \frac{a_{n}}{b_{n}}=0$ and $\sum b_{n}$ diverges, but $\sum a_{n}$ converges.
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