6.4.1

Instructor: Example one, find the binomial series for f of x equals root 1 plus x quantity. And then part b is going to be, use the third-order Maclaurin polynomial, p-3, to estimate the square root of 1.5. And then we'll use Taylor's theorem to bound the error on this. 

So we're going to use the definition that we have here of the Maclaurin series of f of x equals 1 plus x to the r, specifically applied where our r is 0.5, or 1/2 because of our square root. So let's just go ahead and write this here, that is 1 plus x to the 1/2. That's what we're doing. 

All right, so that means that 1 plus x to the 1/2 is going to equal to 1 plus 1/2 x plus, and based off of our definition, 1/2 times, and then we'll subtract 1 from that. So it's negative 1/2, divided by 2 factorial, plus, that will be 1/2 times negative 1/2 times negative 3 over 2 divided by 3 factorial. Oh, I left off my x term here, x squared, x cubed. 

And then I'll continue in [inaudible] subtracting 1 as we go. All right, well, that'll be 1 plus 1/2 x. Now, how could we simplify that? Well, If you notice, we have 1/2, 1/2, it's [inaudible] with a 2 squared there, Minus 1 over 2 squared-- oops, not a factorial-- squared. We have that 2, actually. 

In fact, let's switch that. Let's say that's 1 over 2 factorial, times x squared over 2 squared. After all, that is what it is. OK, this next one's going to be positive because we have a whole bunch of negatives there. A positive, and that's going to be 2 cubed and a 3 factorial. 

All, right so that's x cubed, and that is a 2 to the third. And I feel like I may have missed something, I have a 3 there. OK, so this is the 1 times 3. That one's just 1 times 1. See, it looks like the next term would be negative. And it appears that what's it's going to happen, we'd have a negative and we'd have a 5. 

OK, so the general form for this is going to be-- actually, let's make that a minus, plus. It's going to be negative 1 to the n plus 1, over n factorial. But you have to go pattern hunting with these. n factorial, and then we have x to the-- actually, so we could have 1 times 1. And then we'd have 1 times 3. And then we'd have 1 times 3 times 5, et cetera. 

So I'm going to write this as 1 times 3 times 5 times all the way up to 2 n minus 3. OK, and that's over a power of 2, times x to the n, plus whatever terms come after that. 

The closed form for this-- and this is not real obvious, so I hope you spent some time looking over this to convince yourself of this-- 1 plus the sum. n equals 1, because we're actually including the 0 term as that 1. It doesn't quite fit the formula. That is negative 1, n plus 1, over n factorial, times 1 times 3 times 5 times 2 and minus 3, over 2 to the n, x to the n. 

So convince yourself of that last term that, generic term, and that's the thing you need to figure out. Make sure you agree with that. OK, so part B we want to write the third Taylor polynomial. 

OK, well, based off of that equation we have just there, this would be 1 plus 1/2 x minus 1 over 2 factorial, times 1 over 2 squared, x squared plus 1 over 3 factorial, 1 times 3 over 2 cubed, x cubed. And that will be our final term. 

So that is 1 plus 1/2 x minus, and that would be 4 8, 1/8 x squared plus, let's see, we would end up with 3 over 48, which is 1/8-- no, 1/16. That's [inaudible] 48, 1/16 x cubed. 

Now, we're going to use this polynomial to estimate the square root of 1.5. Well, in this case, if you notice, because our function is the square root of 1 plus x, we actually only need to find p of 0.5. And that is approximately-- let's see, it is-- oh, I don't have that calculated, I thought I did. 

All right, so let's see, that's 1 plus 1/2 times 1/2 minus 1/8 times 0.5 squared plus 1/16 times 0.5 cubed. That is approximately 1.2265625. And the exact square root of 1.5 is 1.224744871. So this is actually pretty good, pretty good estimate. 

Now, for bounding the error, we would find our error on the third remainder, 0.5. That is going to be less than or equal to, based on Taylor's formula, with remainder, m over our x squared, so in our case, 0.5, not squared, but to the fourth power divided by 4 factorial, where M is the maximum value of the fourth derivative, absolute value of fourth derivative on that interval, 0, 0.5. 

Which happens to be at 0.5, that value is equal to 0.9375. So evaluating this for 0.9375, that tells us that our remainder error is approximately 0.02-- no, 0.00244. There's our bound, based on Taylor's theorem of remainder.