Worked example: p-series | Series | AP Calculus BC | Khan Academy

So we have an infinite series here: one plus one over two to the fifth plus one over three to the fifth, and we just keep on going forever. We could write this as the sum from n equals one to infinity of 1 over n to the 5th power, 1 over n to the 5th power.

Now, you might recognize—notice when n is equal to 1, this is 1 over 1 to the 5th; that's that over there. And we could keep on going. Now you might immediately recognize this as a p series, and a p series has the general form of the sum going from n equals 1 to infinity of 1 over n to the p, where p is a positive value.

So, in this particular case, our p for this p series is equal to five; p is equal to five. Now you might already recognize under which conditions for a p series does it converge or diverge. It's going to converge. It's going to converge when your p is greater than one, which is clearly the case in this scenario right over here. Our p is clearly greater than one.

We would diverge; we would diverge if our p is greater than zero and less than or equal or less than or equal to 1. This would be a divergence. So if this was like 0.9 here, or if this was a, you know, three-fourths, then we would be diverging. So at least for this one, we are convergent.

Let's do another one of these. All right, so here you might again recognize this as a p series. Let me rewrite this infinite sum. So this is the sum from n equals 1 to infinity of 1 over—let's see—we have square root of 2, square root of 3. So you could use this as 2 to the one-half, 3 to the one-half, 4 to the one-half. So it's 1 over n to the one half.

Notice this is when n is equal to one: one over one to the one half is one. One over two to the one half, well that's this right over here, and we keep on going on and on and on. Well, in this case, we still have a p series. We have one over n to some power, and that power is positive, but notice in this case our p falls between zero and one.

So one half is our p, so p for our p series is equal to one-half, and that's between zero and one. Remember, we're divergent—divergent when our p is greater than zero and less than or equal to one, which was clearly the case right over here. So this is going to be divergent.