Calculus 2, Fall 2016

## Provocative questions

1. Is it possible to add all of the terms in an infinite sequence and get a number?

2. In order to add an infinite list of nonzero numbers and get a finite result, what must be true about most of the numbers in that list?

3. If we add an infinite list of numbers and the sum diverges (either by oscillating or by approaching $$\pm \infty$$), will the series still diverge if the first 100,000,000 terms are removed?

4. Does the order in which we add infinitely many numbers affect the value of a sum of infinitely many numbers?

## A geometric series

What is the sum of all the terms in the sequence $$\displaystyle \lbrace a_n \rbrace = \left\lbrace \frac{1}{2^n} \right\rbrace$$?

With each jump, Zeno covers half the remaining distance. Does he ever reach his destination?

The series $$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$$ is called geometric because the ratio of successive terms is always the same: $$\frac{a_{n+1}}{a_n} = \frac{1}{2}$$.

## Geometric series

1. Expand $$(1-r)(1+r+r^2+r^3+\cdots+r^n)$$.

2. What is a closed-form expression for $1+r+r^2+r^3+\cdots+r^n?$

3. What is $$\displaystyle \lim_{n \to \infty} r^n$$? What values of $$r$$ make this limit converge?

4. Construct a formula for the geometric series $1+r+r^2+r^3+\cdots$ and extend it to $a+ar+ar^2+ar^3+\cdots$

## Geometric series

1. Consider $$\displaystyle \sum_{n=0}^{\infty} \frac{2}{5^n}$$. Find a closed-form expression for the $$n$$th partial sum $$S_n = 2 + \frac{2}{5} + \frac{2}{25} + \cdots + \frac{2}{5^n}$$. What is the limit of the sequence of partial sums? What is the sum of the series?

2. Is $$7 - \frac{7}{3} + \frac{7}{9} - \frac{7}{27} + \frac{7}{81} - \cdots$$ a geometric series? If yes, what is its ratio? What is the sum of this series?

3. Is $$9 + 3 + 1 + \frac{1}{3} + \frac{1}{9} + \cdots$$ a geometric series? If yes, what is its ratio and what is its sum?

4. Is $$\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots$$ a geometric series? If yes, what is its ratio and what is its sum?

## Bouncing ball

A ball is dropped from a height of 32 feet and bounces. Suppose that each bounce is $$1/4$$ of the height of the bounce before.

1. Find an expression for the height to which the ball rises after it hits the floor for the $$n$$th time.

2. Find an expression for the total vertical distance the ball has traveled when it hits the floor for the first, second, and third times.

3. Find an expression for the total vertical distance the ball has traveled when it hits the floor for the $$n$$th time.

## Square spidron

Assume the largest square has sides of length 1 and that the vertices of each in-square meet the edges of each out-square at the midpoints. If the indicated pattern is continued indefinitely, what will be the area of the shaded region?

## Coffee

An 8 oz cup of coffee has 95 mg of caffeine. The half-life of coffee in the body is approximately 6 hours. You drink one cup of coffee each day at 8 am.

## Coffee

An 8 oz cup of coffee has 95 mg of caffeine. The half-life of coffee in the body is approximately 6 hours. You drink one cup of coffee each day at 8 am.

1. How much caffeine is in your system after one day, just before drinking your morning cup of joe? How much after?

2. How much caffeine is in your system after two days, just before drinking your morning cup of joe? How much after?

3. How much caffeine is in your system after three days, just before drinking your morning cup of joe? How much after?

4. Find closed form expressions for the amount of caffeine in your system after $$n$$ days just before and after you drink your daily coffee.

## Proof without words

Explain how this picture proves the formula for the sum of a geometric series for a ratio $$0 < r < 1$$.

## Telescoping series

1. Find the value of $$\displaystyle \sum_{n=2}^{\infty} \frac{1}{n^2-1}$$

A telescoping series has partial sums that "collapse like a telescope" to a fixed number of terms.

## Divergence test

Divergence test: If $$\displaystyle \lim_{n \to \infty} a_n \ne 0$$, then $$\displaystyle\sum_{n=1}^{\infty} a_n$$ diverges.

Explain why the divergence test is true.

1. Can you construct a sequence such that $$\lbrace a_n \rbrace \to 0$$ as $$n \to \infty$$ and $$\displaystyle\sum_{n=1}^{\infty} a_n$$ converges?

2. Can you construct a sequence such that $$\lbrace a_n \rbrace \to 0$$ as $$n \to \infty$$ and $$\displaystyle\sum_{n=1}^{\infty} a_n$$ diverges?