Date modified: Sat 2023-07-15 . 07:19 PM
Date created: Sat 2023-10-07 . 09:02 PM
# Week 4 Thursday Problems ## Reading. Keep reading chapter 11. We are currently in 11.1 and 11.2. ## Problems. 1. It is important to know how we define convergence of a series. 1. What is the difference between a sequence and a series? 2. What does it mean when we say a series $\sum_{n=1}^{\infty} a_{n}$ converges? 2. Suppose some series $\sum_{n=1}^{\infty} a_{n}$ has an associated partial sum sequence $s_{n}=a_{1}+a_{2}+\cdots +a_{n}$ given by $\displaystyle s_{n}= \frac{n^{2}+1}{3n^{2}-5}$. Does our original series converge or diverge? What does it converge to, if converges? 3. Let consider the series $\displaystyle\sum_{n=1}^{\infty} a_{n}$ given by $\displaystyle a_{n}=\frac{1}{n^{2}}$. Write out explicitly the first 4 terms of the associated partial sum sequence $s_{n}$ for this series. That is, what are $s_{1}, s_{2}, s_{3}$, and $s_{4}$? 4. Consider the sequence $\displaystyle a_{n}= \frac{3n+1}{7n+2}$. 1. Does the sequence $(a_{n})$ converge? If so, what does it converge to? 2. Does the series $\sum_{n=1}^{\infty} a_{n}$ converge? If so, what does it converge to? 5. Determine whether the following series converges or diverges by expressing its partial sum $s_{n}$ as a telescoping sum (see our notes or textbook 11.2). If it is convergent, find its sum. Notice, some you have to write out a few terms to see the pattern, or do some algebraic manipulations first. 1. $\displaystyle\sum_{n=1}^{\infty} \left( \frac{1}{\sqrt{n}}- \frac{1}{\sqrt{n+1}} \right)$ 2. $\displaystyle\sum_{n=1}^{\infty} \left( \frac{1}{n-2} - \frac{1}{n} \right)$ 3. $\displaystyle\sum_{n=1}^{\infty} \frac{3}{n(n+3)}$ 4. $\displaystyle\sum_{n=1}^{\infty} \ln\left( \frac{n}{n+1} \right)$ 6. An important type of series are the geometric series. Determine whether the following geometric series converges or diverges, and if converge, give its sum. Pay attention to the index. (You need to identify what is the first term, and common ratio) 1. $\displaystyle\sum_{n=1}^{\infty} 12(0.73)^{n-1}$ 2. $\displaystyle\sum_{n=5}^{\infty} 5 (0.3)^{n}$ 3. $\displaystyle\sum_{n=1}^{\infty} \frac{e^{2n}}{6^{n-1}}$ 7. Determine whether the following series converge or diverge. 1. $\displaystyle\sum_{n=1}^{\infty} e^{-n}$ 2. $\displaystyle\sum_{n=1}^{\infty} e^{n}$ 3. $\displaystyle\sum_{n=1}^{\infty} \frac{1}{4+e^{-n}}$ 8. Express the following repeating decimals as ratio of integers. 1. $1.\overline{0423}$ 2. $1.04\overline{23}$ 9. Look at the following expressions in $x$ very carefully. They all looks like a geometric series. For what values of $x$ will make those series converge. (Think about what kinds of common ratio we need for a geometric series to converge.) 1. $\displaystyle\sum_{n=1}^{\infty}(x+4)^{n}$ 2. $\displaystyle\sum_{n=1}^{\infty}2^{n}(x-3)^{n}$