Examples of Convergent Series
Today I want to give two examples of convergent series that turn out to be extremely useful for comparisons.
First we have the geometric series whose terms are the sequence for some constant ratio . The sequence of partial sums is
If we can multiply this sum by to find
Then as goes to infinity, this sequence either blows up (for ) or converges to (for ). In the border case we can also see that the sequence of partial sums fails to converge. Thus the geometric series converges if and only if , and we have a nice simple formula telling us the sum.
The other one I want to hit is the so-called -series, whose terms are starting at . Here we use the integral test to see that
so the sum and integral either converge or diverge together. If the integral gives , which converges for and diverges for .
If we get , which diverges. In this case, though, we have a special name for the limit of the difference . We call it “Euler’s constant”, and denote it by . That is, we can write
where is an error term whose magnitude is bounded by .
In general we have no good value for the sums of these series, even where they converge. It takes a bit of doing to find , as Euler did in 1735 (solving the “Basel Problem” that had stood for almost a century), and now we have values for other even natural number values of . The sum is known as Apéry’s constant, after Roger Apéry who showed that it was irrational in 1979. Yes, we didn’t even know whether it was a rational number or not until 30 years ago. We have basically nothing about odd integer values of .
If we say instead of , and let take complex values (no, I haven’t talked about complex numbers yet, but some of you know what they are) we get Riemann’s function , which is connected to some of the deepest outstanding questions in mathematics today.