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# Week 5 Wednesday Problems
## Reading.
Please read sections 11.5 to 11.6. Do try the exercises for additional practice.
We focus now on alternating series test and absolute convergence.
## Problems.
1. Determine whether the following series converge or diverge.
1. $\displaystyle \frac{1}{\ln 3} - \frac{1}{\ln5}+ \frac{1}{\ln 7}-\frac{1}{\ln9}+\cdots$ (Note: Anytime we have the ambiguous notation $\cdots$, continue the pattern in the "sensible way" for our purpose.)
2. $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{3+7n}$
3. $\displaystyle\sum_{n=0}^{\infty} \frac{3n-1}{2n+1}$
4. $\displaystyle\sum_{n=1}^{\infty} (-1)^{n}e^{-n}$
5. $\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{2}}{n^{3}+4}$
6. $\displaystyle\sum_{n=1}^{\infty}(-1)^{n-1} e^{2/n}$
7. $\displaystyle\sum_{n=0}^{\infty}\frac{\sin\left(n\pi+\frac{1}{2}\pi\right)}{1+\sqrt{n}}$
8. $\displaystyle\sum_{n=1}^{\infty} \frac{n\cos(n\pi)}{2^{n}}$
9. $\displaystyle\sum_{n=1}^{\infty}(-1)^{n} \frac{n^{2}}{5^{n}}$
2. Explain what does it mean for a series to be absolutely convergent. And explain what does it mean for a series to be conditionally convergent.
3. Suppose a series $\sum a_{n}$ is such that $\sum|a_{n}|$ converges. What can we saey about the convergence of $\sum a_{n}$
4. Determine whether each of the series below is (1) absolutely convergent, (2) conditionally convergent, or (3) divergent.
1. $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{4}}$
2. $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}$
3. $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt[3]{n^{2}}}$
4. $\displaystyle\sum_{n=1}^{\infty} (-1)^{n} \frac{n^{2}}{n^{2}+1}$
5. $\displaystyle\sum_{n=1}^{\infty} \frac{-n}{n^{2}+1}$
6. $\displaystyle\sum_{n=1}^{\infty} \frac{\sin(n)}{2^{n}}$
7. $\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{n^{2}+4}$
8. $\displaystyle\sum_{n=1}^{\infty}(-1)^{n} \frac{n}{\sqrt{n^{3}+2}}$
9. $\displaystyle\sum_{n=2}^{\infty} \frac{(-1)^{n}}{n\ln(n)}$
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