Date modified: Tue 2023-07-18 . 12:56 AM
Date created: Sat 2023-10-07 . 09:02 PM
# Week 5 Monday Problems ## Reading. Please read sections 11.1 to 11.4. Do try the exercises for additional practice. We focus now on integral test and comparison tests. ## Problems. 1. Use integral test to determine whether the following series converge or diverge. 1. $\displaystyle\sum_{n=1}^{\infty} \frac{1}{(5n+2)^{4}}$ 2. $\displaystyle\sum_{n=3}^{\infty} \frac{5n^{2}}{n^{3}+3}$ 3. $\displaystyle\sum_{n=5}^{\infty} \frac{1}{n(\ln n)^{3}}$ 4. $\displaystyle\sum_{n=2}^{\infty} \frac{\arctan(n)}{1+n^{2}}$ 2. Determine whether the following series converge or diverge. 1. $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{\sqrt{3}}}$ 2. $\displaystyle\sum_{n=1}^{\infty} \frac{\sqrt{n}+5}{n^{3}}$. Hint: Break this series into a sum of two series. If each of the series converges, then the series converges. (If one or both parts diverges, then it's not as clear) 3. $\displaystyle\sum_{n=1}^{\infty}ne^{-n}$ 4. $\displaystyle\sum n e^{-n^{2}}$ 5. $\displaystyle\sum_{n=6}^{\infty} \frac{1}{n(\ln(n))^{3}}$ 6. $\displaystyle\sum_{n=6}^{\infty} \frac{n}{(1+n^{2})^{3}}$ 7. $\displaystyle\sum_{n=6}^{\infty} \frac{\ln(n)}{n^{3}}$ 3. Euler's solved the famous Basel problem by showing $$\sum_{n=1}^{\infty} \frac{1}{n^{2}} = \frac{\pi^{2}}{6}$$ Note this is the sum of the reciprocal of squares. 1. Using above result, find the value of $\displaystyle\sum_{n=1}^{\infty} \frac{1}{(2n)^{2}}$. This is the sum of the reciprocal of squares of even numbers. 2. Try using above results further, and deduce the value of $\displaystyle\sum_{n=1}^{\infty} \frac{1}{(2n-1)^{2}}$. What is this summing? (Hint: Write out a few terms of these series out to give you an idea.) 4. Use direct comparison test to determine whether the following series converge or diverge. 1. $\displaystyle\sum_{n=1}^{\infty} \frac{n}{n^{5}+3n+1}$ 2. $\displaystyle\sum_{n=3}^{\infty} \frac{n^{2}+n}{n^{3}-2}$ 5. Review all the tests and known series that you have so far. ////