Harmonic series and p-series
Worked example: p-series
- [Instructor] 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 one over n to the fifth power, one over n to the fifth power. And now you might recognize, notice, when n is equal to one, this is one over one to the fifth, 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 one to infinity, of one 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 one. This would be a divergent. So if this was like .9 here, or if this was a, you know, 3/4, 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 one to infinity of one over, let's see, we have a square root of two, a square root of three. So you could do this as two to the 1/2, three to the 1/2, four to the 1/2. So it's one over n to the 1/2. Notice, this is when n is equal to one. One over one to the 1/2 is one. One over two to the 1/2, 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 1/2 is our p. So p for our p-series is equal to 1/2, 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.
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