"We now know that when students make a mistake in math class, their brain grows, synapses fire, and connections are made. This finding tells us that we want students to make mistakes in math class and that students should not view mistakes as learning failures but learning achievements. But students everywhere think this means they are not a 'math person.' We need to change this thinking by telling students that mistakes are productive."

– Jo Boaler (What's Math Got to Do With It? p. xviii-xix)

Theorem: Suppose \(a(x)\) is a continuous, positive, decreasing function on \(x \geq 1\). Define a sequence \(\lbrace a_n \rbrace\) by \(a_n = a(n)\) for every natural number \(n\). Then the series \(\displaystyle\sum_{n=1}^{\infty} a_n\) converges if and only if the integral \(\displaystyle\int_{x=1}^{\infty} a(x) \, dx\) converges.

The left sum represents \(\displaystyle \sum_{n=1}^{\infty} a_n\).

Choose converges or diverges to make true statements.

If \(\int_{1}^{\infty} a(x) \, dx\) ( converges / diverges ),
then \(\sum_{n=1}^{\infty} a_n\) ( converges / diverges ).

If \(\sum_{n=1}^{\infty} a_n\) ( converges / diverges ),
then \(\int_{1}^{\infty} a(x) \, dx\) ( converges / diverges ).

The right sum represents \(\displaystyle \sum_{n=2}^{\infty} a_n\).

If \(\sum_{n=2}^{\infty} a_n\) ( converges / diverges ),
then \(\int_{1}^{\infty} a(x) \, dx\) ( converges / diverges ).

If \(\int_{1}^{\infty} a(x) \, dx\) ( converges / diverges ),
then \(\sum_{n=2}^{\infty} a_n\) ( converges / diverges ).

If \(\sum_{n=2}^{\infty} a_n\) ( converges / diverges ),
then \(a_1 + \sum_{n=2}^{\infty} a_n = \sum_{n=1}^{\infty} a_n\) ( converges / diverges ).

1. Does \(\displaystyle \sum_{n=1}^{\infty} \frac{1}{n}\) converge? Check the validity conditions!

2. For what values of \(p\) does \(\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^p}\) converge? Explain your answer.

3. Does \(\displaystyle \sum_{n=1}^{\infty} \frac{n}{2^n}\) converge? Check the validity conditions!

Theorem: Suppose \(0 \leq a_n \leq b_n \leq c_n\) for all \(n\) greater than some (fixed) integer \(N \geq 1\).

1. If \(\displaystyle \sum_{n=1}^{\infty} a_n\) diverges, then \(\displaystyle \sum_{n=1}^{\infty} b_n\) diverges.

2. If \(\displaystyle \sum_{n=1}^{\infty} c_n\) converges, then \(\displaystyle \sum_{n=1}^{\infty} b_n\) converges.

Restate these in your own words plain English and without using symbols: "For positive sequences…"

Make an informed guess: the series ( converges / diverges ), so we should try to bound it ( above / below ) by something that ( converges / diverges ). Find a bounding function and show your guess is right!

1. \(\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2+3n-5}\).

2. \(\displaystyle \sum_{n=1}^{\infty} \frac{1}{n! + n}\).

3. \(\displaystyle \sum_{n=4}^{\infty} \frac{n^2+n+1}{n^3-5}\).

1. Show \(\displaystyle \sum_{n=1}^{\infty} \frac{1}{n}\) diverges (by grouping and bounding).

2. If a sequence \(a_n\) is bounded above by \(\displaystyle\frac{1}{n}\), must \(\displaystyle\sum_{n=1}^{\infty} a_n\) converge? Does \(\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{1.00000001}}\) converge?

3. Is \(\displaystyle a_n = \frac{1}{n\ln(n)}\), \(n \geq 2\), bounded above by \(\displaystyle \frac{1}{n}\)?

4. Does \(\displaystyle \sum_{n=2}^{\infty} \frac{1}{n\ln(n)}\) converge or diverge? Explain.

Does this picture show that \(\displaystyle \sum_{n=2}^{\infty} \frac{1}{n^{1.01}}\) converges but \(\displaystyle \sum_{n=2}^{\infty} \frac{1}{n \ln(n)}\) diverges? Why or why not? What must happen?

1. How does \(1,\ 0.5,\ 0.25,\ 0.125,\ 0.0625,\ 0.03125, \ldots\) compare to \(4,\ 2,\ 1,\ 0.5,\ 0.25,\ 0.0125, \ldots\) in the long run?

2. Will \(4 + 2 + 1 + 0.5 + 0.25 + 0.0125 + \cdots\) converge? Why?

3. How does \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \ldots\) compare to \(1.1, \frac{1}{1.9}, \frac{1}{3.1}, \frac{1}{3.9}, \frac{1}{5.1}, \ldots\) in the long run?

4. Will \(1.1 + \frac{1}{1.9} + \frac{1}{3.1} + \frac{1}{3.9} + \frac{1}{5.1} + \cdots\) converge? Why?

Theorem: Let \(\lbrace a_n \rbrace\) and \(\lbrace b_n \rbrace\) be positive sequences.

1. If \(\displaystyle \lim_{n \to \infty} \frac{a_n}{b_n} = L\) for some positive number \(L\), then \(\displaystyle\sum_{n=1}^{\infty} a_n\) and \(\displaystyle\sum_{n=1}^{\infty} b_n\) either both converge or both diverge.

2. If \(\displaystyle \lim_{n \to \infty} \frac{a_n}{b_n} = 0\) and \(\displaystyle\sum_{n=1}^{\infty} b_n\) converges, then \(\displaystyle\sum_{n=1}^{\infty} a_n\) converges.

3. If \(\displaystyle \lim_{n \to \infty} \frac{a_n}{b_n} = \infty\) and \(\displaystyle\sum_{n=1}^{\infty} b_n\) diverges, then \(\displaystyle\sum_{n=1}^{\infty} a_n\) diverges.

Make an informed guess: the series \(\sum a_n\) ( converges / diverges ), so we should try to find a simpler series \(\sum b_n\) that ( converges / diverges ) and determine if the ratio \(\frac{a_n}{b_n}\) approaches a constant as \(n \to \infty\) or diverges to \(\infty\).

1. \(\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2-3n+5}\).

2. \(\displaystyle \sum_{n=1}^{\infty} \frac{e}{\sqrt{n^2+n}}\).

3. \(\displaystyle \sum_{n=1}^{\infty} \frac{\ln(n)}{n!}\).