6.2.4

Instructor: Use the power series representation of f of x equals 1 over 1 minus x-- we know that very well-- to find a power series representation for g of x, which is 1 over 1 minus x quantity squared. Notice that g of x equals f prime of x. So if we look at the derivative of our power series, I'm going to go in and write it out term by term. It is 1 plus x plus x squared plus x cubed plus something else. 

Now, if I take the derivative, that is 0-- I'm going to go and include that-- plus 1 plus 2x plus 3x squared plus-- so right there. Now what I want to do is write this in a closed form. And it may take some thinking for you to recognize what this is. But there is a pattern with the constants. We do have x to the n. 

But we're going to have an n plus 1 there. So if you plug in 0, that's 1 with x to the 0, so that's just the 1 term right there. If you plug in a value of 1, that is 2x to the first. Value of 3, that is 4. Wait, hold on. I said 2. Oh, that's 3. 3x squared, plug in a value of 3, that's 4x to the 4-- or 3. 

So we have that building up there. So recognize the pattern. It's going-- and that's the closed form right there. Now, one question we have is, determine the interval of convergence. Well, if our derivative-- see, if our original function had an interval of a minus r, a plus r-- that's where it's centered, a-- then our derivative is going to have the same interval of convergence. 

So this means-- leave that underlined. That means that this is going to have an interval of convergence of absolute value of x is less than 1. Now let's actually look at the endpoints. So that would be a negative 1 to 1. 

So if x is negative 1 if x is negative 1, then that makes our interval n plus 1 to the negative 1 to the n. OK, now that is going to diverge because it's an alternating series, where this is not decreasing. So that diverges. 

All right, what if x equals 1? Well, then that would be the sum of n plus 1. And that would be it. So that also diverges. So our interval of convergence is negative 1 to 1. And there is our power series representation for 1 minus x quantity squared. 

Now let's use that to evaluate the sum of this series. All we really need to notice is that this series we're given is not a power series. It is just a series. And we can write this is n plus 1 times 1/4 to the n, which means x is 1/4. So this is actually the same as g of 1/4, which is 1 over 1 minus x quantity squared. 1 over 1 minus 1/4 squared. All right, 3/4 squared, that's 9/16. So 16/9 is the sum of that series. 

[ENDS AT 4:16]