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Tuesday, June 14, 2005

Infinite Series: integral test

Ok, let's develop an important tool: this is called the integral test.

First of all, remember what a Riemann Sum is; in particular recall the left and right endpoint sums. (Recall all of those little rectangles that you drew when you were first learning about definite integrals).

Next, recall what an improper integral is. This is where you take a limit as one of the limits of the definite integral goes to infinity.

Now suppose f is a function which, on the interval [1, infinity) is decreasing and positive (examples: f = 1/x, f = exp(-x) ). We then have the following:

the sum f(1)+ f(2) +.....+ f(k) + f(k+1) +..... converges if and only if the improper integral
integral(f(x), 1, infinity) converges.

Why? Note that f(1)+ f(2) + .....+f(k) + f(k+1)+...... is a left hand endpoint sum for the integral and therefore is larger than the integral. (why? draw a picture and draw the rectangles for the left hand sum). Note that f(2)+...+f(k) + f(k+1)+.... is a right hand endpoint sum for the improper integral and is smaller than the integral.

Also remember that both sums have a sequence of partial sums which are increasing.

So, if the sum is finite, so is the integral. If the sum is infinite, then so is the sum starting with
f(2) and so is the integral.

That is the integral test, in a nutshell.

As a consequence: sums whose terms look like 1/(ak+b) where a and b are constants are infinite and sums whose terms look like 1/p(k) where p(k) is a polynomial of degree 2 or larger converge.


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