Date modified: Sat 2023-07-15 . 06:38 PM
Date created: Sat 2023-10-07 . 09:02 PM
# Week 4 Wednesday Problems ## Reading. Start reading chapter 11. Start with 11.1 and 11.2. ## Problems. 1. Determine if the each of the following sequences converges or diverges. If converges, find the limit of the sequence. 1. $\displaystyle a_{n}=\frac{5}{n+2}$ 2. $a_{n}=5\sqrt{n}$ 3. $\displaystyle a_{n} = \frac{4n^{2}-3n}{2n^{2}+1}$ 4. $\displaystyle a_{n}=\frac{n^{4}}{n^{3}-3n}$ 5. $a_{n}=4 - (0.87)^{n}$ 6. $a_{n}=3^{n}\cdot 4^{-n}$ 7. $\displaystyle a_{n}=\frac{\sqrt{5n}}{\sqrt{n}+\sqrt{2}}$ 8. $\displaystyle a_{n}=\left( 1+\frac{5}{n} \right)^{n}$ . Hint: L'Hospital rule or recognize it from somewhere. 9. $\displaystyle a_{n}=n^{1/n}$ 10. $\displaystyle a_{n}=n \sin\left( \frac{1}{n} \right)$ 11. $a_{n}=\ln(2n^{2}+1)-\ln(n^{2}+1)$ 12. $\displaystyle a_{n}=\frac{n!}{2^{n}}$. Think about all the terms in $n!$ and all the terms in $2^{n}$ 2. Determine if the following sequences is monotonic. Also determine if it is bounded. 1. $a_{n}=\cos(n)$ 2. $\displaystyle a_{n}=\frac{1}{2+3n}$ 3. Consider the sequence $(a_{n})$ given by $a_{1}=\sqrt{2}$ and $a_{n+1}=\sqrt{2+a_{n}}$. 1. Show the sequence $(a_{n})$ converges. 2. Find the limit of this sequence. 4. Consider the sequence $(a_{n})$ given by $a_{1}=1$ and $\displaystyle a_{n+1}=3-\frac{1}{a_{n}}$. 1. Show this sequence $(a_{n})$ converges. 2. Find the limit of this sequence. 5. Consider the sequence $(a_{n})$ given by $a_{1}= 2$ and $\displaystyle a_{n+1}=\frac{1}{3-a_{n}}$. 1. Show this sequence $(a_{n})$ converges. 2. Find the limit of this sequence. ///