2007-10-18 - The Integral Test
Today's nonsense
- Seinfeld reference: we're all laughing at the little doodle from physics, but nobody seems to know what it's about.
- "Are you kidding me?" - Phoebe's senior quote. Officially.
- "Shift Happens" shirt v. "lie tangent to the curves shirt"
- Evidently I have to send an email out to Math club and thensome to Physics
If f is positive, continuous, and decreasing for x >= 1 and a_n=f(x), then the series that we are questioning (whether it diverges / converges) will do the exact same thing as the integral of the function does from 1 to infinity. So whatever the integral does, if you were to integrate from n=1 to n=infinity, whatever the integral does, the series does.
ex 1) Use the integral test to solve the sum from n=1 to infinity of n / (n^2 + 1)
-- Use u-substitution to solve the equivalent integral.
ex 2) sum from n=1 to infinity of 1 / (n^2 + 1)
-- This is an arctangent. Arctangent(infinity) is pi/2 and arctan(1) is pi/4 so the integral converges to pi/4 but we just need to know that the series/integral here converges.
All of these are playing off with what we did with integrals. So now we are going to talk about p-series.
P-series and Harmonic Series
We've mentioned this before, maybe not calling it P, but we've used the letter P.
The series from n=1 to infinity of 1/(n^p)
If we are going to infinity, if p is a certain number, the exponent on the variable in the denominator, well, it's going to converge with certain numbers. If p>1 it converges.
1. The p-series converges if p>1. Otherwise it's a natural log, and ln diverges.
2. The p-series diverges if 0 < p=<1. If p is negative, then it just flips back on top, and it diverges. So if it's negative, it's not really in the denominator.
The harmonic series is a special p-series. What possible special series could we have? One. Harmonic series is when p=1. So,
Series from n=1 to infinity of 1/n, and you know it converges to zero. So a harmonic series is a p-series, and it's when p=1. Harmonic tones are those with strings that have a certain length, and when played maybe their amplitudes all add up, musically? Whatever.
We know it converges both by the integral test and p-series. A general harmonic series is of the form:
series 1/(an+b)
The capital pi symbol is useful for multiplication, instead of the sigma symbol. Useful in combinations and permutations.
"Or a capital H with the top cut off." - Dakota.
Coffee table book about coffee tables. And then ether on the road, and then Numan's on fire, sleeping 15 minutes every 4 hours, ... Seinfeld reference overload.
ex 1) Sigma from n=1 to infinity of 1/n^2.
-- p series test because p>1. Or the integral test. Converges.
ex 2) Sigma from n=1 to infinity of 1 / (n ln(n) )
-- Diverges. Use u-substitution.
Homework due tomorrow.
17 min.
Concise statement of the integral test
Wikipedia: Integral test for convergence
Wolfram: Integral test
Math Refresher blog entry/article
Paul's notes on the integral test
Wikipedia: Harmonic series
Rearranging the alternating harmonic series (hey, look- some actual writing)
** "A Sweet Introduction to Series" (Dale Hoffman (no, not the other guy))