1. So we have the definition for a series to converge, but what about some slick tests?
Wait a moment: wasn’t a Riemann Sum an infinite series? Why don’t we use integration to test if a series converges or diverges?
We can do something like this. Lets investigate it further.
2. An infinite series could be thought of as where is a function of a real number…we just restrict attention to positive integers.
We can obviously test if as using L’Hopital’s Rule. So we will assume that for each .
So what? Well, we see that
It thus stands to reason that we can sum over to get
Consequently, the series converges if and only if the integral is finite.
If the integral is infinite, the series diverges. This is quite clean, clever, and cute.
Example 1 (Harmonic Series). The classic example is the Harmonic series
We see that the function has its integral be
Taking the limit as gives us . By the integral test, the Harmonic Series diverges!
3. Usefulness. Consider the series
What does it converge to? Does it converge? Well, we see that the integral test gives us
Thus we see the series converges.
Moreover, we can use the integral test to tell us how good using, e.g., the first eight terms would estimate the sum. We see that
So using only the first eight terms or so gives us an approximation good to a digit or so. But that’s something!
Theorem 1. The series , where is a fixed constant, converges if and only if
Proof. We use the integral test on . We see if , the integral test tells us the series diverges:
On the other hand, we see if , we have the integral test tell us
which implies the series converges. ▮