Section 11.1

Key Problems: 21, 22, 29, 34, 38, 43, 49, 51, 65, 78, 79

Problem 21

Find a formula for the general term $a_n$ of the sequence, assuming that the pattern of the first few terms continues.

$$\left\{\frac{1}{2},-\frac{4}{3},\frac{9}{4},-\frac{16}{5},\frac{25}{6},\dots \right\}$$

Answer

Note that the sequence can be written as shown below:

$$\begin{array}{rl}{a}_1& =(-1{)}^{1+1}\times \frac{1^2}{1+1}=\frac{1}{2}\\ a_2& =(-1{)}^{2+1}\times \frac{2^2}{2+1}=-\frac{4}{3}\\ a_3& =(-1{)}^{3+1}\times \frac{3^2}{3+1}=\frac{9}{4}\\ a_4& =(-1{)}^{4+1}\times \frac{4^2}{4+1}=-\frac{16}{5}\\ a_5& =(-1{)}^{5+1}\times \frac{5^2}{5+1}=\frac{25}{6}\end{array}$$

So we can define the formula as:

$$a_n=(-1{)}^{n+1}\times \frac{n^2}{n+1}$$

Problem 22

$$\left\{1,0,-1,0,1,0,-1,0,\dots \right\}$$

Problem 29

Determine whether the sequence converges or diverges. If it converges, find the limit.

$$a_n=\frac{4n^2-3n}{2n^2+1}$$

Problem 34

$$a_n=\frac{3\sqrt{n}}{\sqrt{n}+2}$$

Problem 38

$$a_n=\cos \left(\frac{n\pi }{n+1}\right)$$

Problem 43

$$\left\{\frac{(2n-1)!}{(2n+1)!}\right\}$$

Problem 49

$$a_n=\frac{{\cos }^{2}n}{2^n}$$

Problem 51

$$a_n=n\sin \left(\frac{1}{n}\right)$$

Problem 65

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess.

$$a_n=\arctan \left(\frac{n^2}{n^2+4}\right)$$

Problem 78

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?

$$a_n=\cos n$$

Problem 79

$$a_n=\frac{1}{2n+3}$$