# Section 5.1 Calculus 2

Sequences

## 5.1 Sequences

### 5.1.1 Definition

• A sequence is an infinitely long list of real numbers. For example, the sequence of positive even integers is $$\<2,4,6,8,\dots\>$$.
• Example Use your intuition to guess the next three terms of the sequences $$\<1,3,5,7,9,\dots\>$$, $$\<3,-6,9,-12,15,\dots\>$$, and $$\<0,1,4,9,16,\dots\>$$.
• An explicit formula $$a_n$$ is a rule defining each term of the sequence, where $$n=0$$ yields the first term, $$n=1$$ gives the next term, and so on. The sequence generated by the formula $$a_n$$ is written as $$\<a_n\>_{n=0}^\infty=\<a_0,a_1,a_2,\dots\>$$.
• Occasionally the first term of the sequence may be given by an integer different from $$0$$, in which case the sequence is written like $$\<a_n\>_{n=1}^\infty$$.
• Example Write the first five terms of the sequences $$\<a_n\>_{n=0}^\infty$$, $$\<b_n\>_{n=0}^\infty$$, and $$\<c_n\>_{n=0}^\infty$$ defined by $$a_n=4n$$, $$b_n=\frac{(-1)^n}{n^2+2}$$, and $$c_n=\cos(\frac{\pi}{2}n)$$.
• Example Give the term $$a_7$$ for the sequence defined by the formula $$a_n=\frac{n}{2n+1}$$.

### 5.1.2 Recursive Formulas

• A recursive formula for a sequence defines one or more initial terms of the sequence, and then defines future terms of the sequence by using previous terms.
• Example Write the first ten terms of the Fibonacci sequence defined by the recursive formula $$f_0=1,f_1=1,f_{n+2}=f_n+f_{n+1}$$.
• Example Write the first six terms of the factorial sequence defined by the recursive formula $$!_0=1,!_{n+1}=(n+1)!_n$$.
• The factorial sequence is commonly written in the form $$n!$$ rather than $$!_n$$. It has the explicit formula $$n!=1\times2\times3\times\dots\times n$$.
• Example Prove that $$a_n=\frac{3}{2^n}$$ is an explicit formula for the sequence $$\<a_n\>_{n=0}^\infty$$ defined recursively by $$a_0=3,a_{n+1}=\frac{a_n}{2}$$.

### 5.1.3 Limits, Convergence, and Divergence

• The sequence $$\<a_n\>_{n=i}^\infty$$ converges to a limit $$L$$ if for each $$\epsilon>0$$, there exists an integer $$N$$ such that $$|a_n-L|<\epsilon$$ for all $$n\geq N$$. This is written as $$\lim_{n\to\infty}a_n=L$$ or $$a_n\to L$$.
• Example Guess the limit of the harmonic sequence
$$\<a_n\>_{n=1}^\infty$$ defined by $$a_n=\frac{1}{n}$$ by writing out the first few terms.
• Example Guess the limit of the sequence
defined by $$g_n=\frac{2^n}{2^{n+1}}$$ by writing out the first few terms.
• A sequence diverges when it doesn’t converge to any limit.
• Example Write a few terms of the sequence defined by the formula $$b_n=(-1)^n\frac{n+1}{n+2}$$. Does it appear to be converging or diverging?

### Review Exercises

1. Use your intuition to guess the next three terms of the sequences $$\<1,5,9,13,17,\dots\>$$, $$\<1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\frac{1}{25},\dots\>$$, and $$\<\frac{1}{3},-1,3,-9,27,\dots\>$$.
2. Create an explicit formula for each of the three previous sequences.
3. Write the first five terms of the sequences $$\<a_n\>_{n=0}^\infty$$, $$\<b_n\>_{n=0}^\infty$$, and $$\<c_n\>_{n=0}^\infty$$ defined by $$a_n=3n+2$$, $$b_n=2(-\frac{1}{3})^n$$, and $$c_n=\frac{n}{1+n^2}$$.
4. Write the first six terms of the sequence $$\<q_n\>_{n=0}^\infty$$ defined by $$q_0=0$$ and $$q_{n+1}=q_n+2n+1$$.
5. Prove that $$q_n=n^2$$ is an explicit formula for the sequence defined recursively in the previous problem.
6. Write the first six terms of the sequence $$\<b_n\>_{n=1}^\infty$$ defined by $$b_1=4$$ and $$b_{n+1}=\frac{b_n}{2}$$.
7. Prove that $$b_n=\frac{8}{2^n}$$ is an explicit formula for the sequence defined recursively in the previous problem.
8. Guess the limit of the alternating harmonic sequence
$$\<b_n\>_{n=1}^\infty$$ defined by $$b_n=\frac{(-1)^n}{n}$$ by writing out the first few terms.
9. Guess the limit of the geometric sequence
$$\<g_n\>_{n=0}^\infty$$ defined by $$g_n=2^{-n}$$ by writing out the first few terms.
10. Guess the limit of the sequence
$$\<a_n\>_{n=3}^\infty$$ defined by $$a_n=\frac{3n+2}{2n+1}$$ by writing out the first few terms.
11. Write a few terms of the sequence defined by the formula $$c_n=\frac{n!}{n^2+1}$$. Does it appear to be converging or diverging?
12. Write a few terms of the sequence defined by the formula $$s_n=\sin(\frac{\pi n}{3})$$. Does it appear to be converging or diverging?
13. Sketch a picture which explains why $$\lim_{n\to\infty} \sin(\pi n)=0$$ as the limit of a sequence, but $$\lim_{x\to\infty}\sin(\pi x)$$ does not exist as a limit of a function.
14. What are the first five terms of the sequence $$\<r_n\>_{n=1}^\infty$$ defined explicitly by $$r_n=\frac{n+2}{3+n^2}$$?
15. What are the first five terms of the sequence $$\<w_n\>_{n=0}^\infty$$ defined recursively by $$w_0=1$$, $$w_1=2$$, $$w_{n+2}=2w_n+w_{n+1}$$?
16. Which of these statements seems most appropriate for describing the sequence whose initial terms are $$\<1,\frac{3}{4},\frac{5}{8},\frac{9}{16},\frac{17}{32},\dots\>$$?
• The sequence appears to converge to $$\frac{1}{2}$$.
• The sequence appears to diverge to $$\frac{1}{2}$$.
• The sequence appears to neither converge nor diverge.

Solutions 1-7

Solutions 8-16

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 9.1