Understanding the p - series test

追加: 2026-05-07

[00:00:00]Let's talk about the P-series test. This is going to be in the form of the sum from K equals 1 to infinity of 1 over K to the power of P. Now, we want to know for what values of P is this going to converge or diverge? Let's go ahead and apply the integral test. Here we have our function f of x equals 1 over x to the power of P and our interval 1 to infinity. We know to apply this test it needs to be continuous, positive, and decreasing, which it is. So, let's go ahead and use it. I'm first going to write 1 over x to the power of P as x to the negative P. That way I can apply the power rule really nicely. We get x to the negative P plus 1 divided by negative P plus 1. Now, I'm just going to go ahead and drop that down to the denominator, so we get 1 over negative P plus 1 times x to the P minus 1. Now, 1 over negative P plus 1 doesn't have anything to do with our limit cuz it doesn't have any values of B, so I can just go ahead and pull it out. Now, I'm going to plug in upper minus lower. We have the limit as B approaches infinity of 1 over B to the power of P minus 1 minus 1 divided by 1 to the power of P minus 1. But, 1 raised to any power is just going to always be 1, so really we're just subtracting 1.

[00:01:14]Now, let's examine what's happening within this limit. I know as B approaches infinity 1 over B to the power of something is going to approach zero. As B gets really, really big, 1 over the big number gets really, really small.

[00:01:29]But, there's an issue if this moves up to the numerator. As B approaches infinity, B to any power is just going to get really, really big and approach infinity. So, if I want B to stay in the denominator, then I need the power P minus 1 to be greater than zero. This tells us that P needs to be greater than 1 for this series to converge.

[00:01:50]Otherwise, if P is less than or equal to 1, this series diverges.