Calculus 2, Fall 2016

## Sequences

Definition: A sequence is an ordered list of infinitely many numbers indexed by the natural numbers $$1, 2, 3, \ldots$$

Equivalently, a sequence is a function $$a(n)$$ whose domain is the natural numbers $$\mathbb{N}$$. We usually write $$a_n$$ for the $$n$$th term in the sequence and use $$\lbrace a_n \rbrace$$ to represent the entire sequence.

$$1, 4, 9, 16, 25, \ldots$$

$$-1, +1, -1, +1, -1, \ldots$$

$$1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \ldots$$

$$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots$$

## Prelude to convergence (numerically)

Consider the sequence $$\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots$$

1. Find a formula for the $$n$$th term, $$a_n$$, of this sequence.
2. What value are the terms in the sequence approaching as $$n \to \infty$$? Does any term in the sequence equal this value?
3. Which terms in the sequence are between $$0.9$$ and $$1.1$$?

## Prelude to convergence (analytically)

Consider the sequence $$\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots$$

1. Which terms in the sequence are between $$0.99$$ and $$1.01$$?
2. Which terms in the sequence are between $$1-\epsilon$$ and $$1+\epsilon$$, where $$\epsilon$$ is an arbitrary positive number? That is, solve the inequalities $$1-\epsilon < a_k$$ and $$a_k < 1+\epsilon$$ for $$k$$.
3. According to your formula, when $$\epsilon = 0.00027$$, what is the smallest value of $$k$$ that will guarantee the distance between $$a_n$$ and $$1$$ is less than $$\epsilon$$ for all $$n > k$$?

## Convergence to a limit

Informal definition: We say that a sequence $$\lbrace a_n \rbrace$$ has limit $$L$$ if the distance between $$a_n$$ and $$L$$ is smaller than any arbitrary positive number for all $$n$$ that are sufficiently large.

Formal definition: Suppose $$\lbrace a_n \rbrace$$ is a sequence and $$L$$ is a real number. We say that $$L$$ is the limit of the sequence as $$n$$ goes to infinity if for every positive number $$\epsilon$$ there can be found some positive integer $$k$$ so that for any $$n$$ greater than $$k$$, the distance between $$a_n$$ and $$L$$ is less than $$\epsilon$$. In other words:

$$L$$ is the limit of $$\lbrace a_n \rbrace$$ if
for all $$\epsilon > 0$$
there exists an integer $$k$$
so that for any $$n > k,$$
$$|a_n - L| < \epsilon$$.

## Convergence to a limit

Note: $$a(25) = 2.1041369$$ and $$a(28) = 1.9065804$$

## Example

Consider $$\lbrace a_n \rbrace = \lbrace \cos(n) \rbrace$$.

1. Is this sequence convergent or divergent?
2. Is this sequence unbounded, bounded above, below, or both? If they exist, give examples of bounds.
3. Is this sequence monotonically increasing or decreasing?
4. Does every bounded sequence converge?
5. If it converges, what does it converge to (i.e., what is the limit)? If it diverges, explain how the formula could be modified to create a convergent sequence.

## Examples

1. $$\displaystyle a_n = \frac{3n}{\sqrt{n^2+1}}$$
2. $$\displaystyle b_n = (-1)^n \cdot \frac{3n-1}{n}$$
3. $$\displaystyle c_n = \frac{n}{n+1} - \frac{n+1}{n}$$
4. $$\displaystyle d_n = \frac{1.1^n}{n}$$
5. $$\displaystyle f_n = (-1)^{n+1} \cdot \frac{n^2}{2^n}$$
6. $$\displaystyle g_n = \frac{\ln(n^2)}{n^2}$$

## Example $$a(n) = \frac{3n}{\sqrt{n^2+1}}$$

L'Hospital's rule applies to the differentiable function $$f(x)$$, not the discrete function $$a(n)$$. Why? Is $$a(n)$$ bounded? Monotonic? Convergent? $$a_n \to$$ _____ as $$n \to \infty$$. Hint: use algebra!

## Example $$b(n) = (-1)^n \frac{3n-1}{n}$$

1. Is $$b(n)$$ bounded? What are lower and upper bounds (numbers)? Is $$b(n)$$ bounded above and below by functions?
2. Is $$b(n)$$ monotonic? How can you tell just by looking at the formula?
3. Is $$b(n)$$ convergent or divergent? What is $$\displaystyle \lim_{n \to \infty} b_n$$?

## Example $$c(n) = \frac{\ln(n)}{n}$$

1. Is $$c(n)$$ bounded? What are lower and upper bounds (numbers)? How can you find the max? Hint: use calc 1!
2. Is $$c(n)$$ monotonic? How can you tell just by looking at the formula?
3. Is $$c(n)$$ convergent or divergent? What is $$\displaystyle \lim_{n \to \infty} c_n$$?

## Example of a common limit

What is a formula for each sequence? What is the limit of each sequence?

1. $$1,1,1,1,\ldots$$
2. $$0.9, 0.99, 0.999, 0.9999, \ldots$$
3. $$1.1, 1.01, 1.001, 1.0001, \ldots$$
4. $$0.9, 1.01, 0.999, 1.0001, \ldots$$

Does $$1 = 0.99999999\ldots$$?

## Example

Consider $$\displaystyle \lbrace a_n \rbrace = \left\lbrace \frac{3^n}{n!} \right\rbrace$$.

1. Find the values of the first 6 terms in this sequence and plot them.
2. Is this sequence convergent or divergent?
3. Is this sequence unbounded, bounded above, below, or both? If they exist, give examples of bounds.
4. Is this sequence monotonically increasing or decreasing?
5. Does every bounded sequence converge?
6. If it converges, what does it converge to (i.e., what is the limit)?

## Comparing growth rates

What kinds of functions (polynomials, exponential, factorial, logarithmic, power) grow faster than others? Evaluate and compare the following limits, and make your own examples.

1. $$\displaystyle \lim_{n \to \infty} \frac{2^n}{n^{100}}$$ and $$\displaystyle \lim_{n \to \infty} \frac{n^{100}}{2^n}$$
2. $$\displaystyle \lim_{n \to \infty} \frac{2^n}{3^n}$$ and $$\displaystyle \lim_{n \to \infty} \frac{3^n}{2^n}$$
3. $$\displaystyle \lim_{n \to \infty} \frac{2^n}{\ln(n)}$$ and $$\displaystyle \lim_{n \to \infty} \frac{\ln(n)}{2^n}$$
4. $$\displaystyle \lim_{n \to \infty} \frac{2^n}{n!}$$ and $$\displaystyle \lim_{n \to \infty} \frac{n!}{2^n}$$
5. $$\displaystyle \lim_{n \to \infty} \frac{2^n}{\sqrt{n}}$$ and $$\displaystyle \lim_{n \to \infty} \frac{\sqrt{n}}{2^n}$$

## Monotonic sequences

A sequence $$\lbrace a_n \rbrace$$ is known to be monotonic increasing.

1. According to the definition of monotonic increasing, what property must this sequence have? What does the graph of a monotonic increasing sequence look like?

2. A monotonic increasing sequence ( must / may / cannot ) have an upper bound.

3. A monotonic increasing sequence ( must / may / cannot ) have a lower bound.

Give an example to show each possibility or impossibility. How must your answers be revised if the same questions were asked about an arbitrary monotonic decreasing sequence?

## Bounded sequences

Suppose $$\lbrace a_n \rbrace$$ is known to be a bounded sequence.

1. A bounded sequence ( must / may / cannot ) be a convergent sequence.

2. A bounded sequence ( must / may / cannot ) be monotonic.

Give an example to show each possibility or impossibility and explain your answer.

## Alternating sequences

Suppose $$\lbrace a_n \rbrace$$ is known to be an alternating sequence (i.e., the signs of the terms alternate between $$+$$ and $$-$$).

1. An alternating sequence ( must / may / cannot ) converge.

2. An alternating sequence ( must / may / cannot ) be bounded.

Give an example to show each possibility or impossibility and explain your answer.

## Unbounded sequences

A sequence $$\lbrace a_n \rbrace$$ is known not to be bounded above.

1. A sequence that is not bounded above (must / may / cannot) contain a positive term.

2. A sequence that is not bounded above (must / may / cannot) have infinitely many positive terms.

3. A sequence that is not bounded above (must / may / cannot) be bounded below.

Give an example to show each possibility or impossibility and explain your answer.

## Convergent sequences

A sequence $$\lbrace a_n \rbrace$$ is known to converge.

1. A convergent sequence ( must / may / cannot ) be monotonic.

2. A convergent sequence ( must / may / cannot ) be bounded.