00:01
All right, so you might know that the formula for a mclaurin series is, in general, it's f of x is equal to the sum as n goes from zero to infinity.
00:15
It's a zero right here.
00:20
There we go.
00:21
Of the nth derivative evaluated at zero times x to the n power divided by n factorial.
00:32
And so the challenge in finding a mclaren series is usually calculating all those derivatives and evaluating them at zero, which can take forever, assuming you can even find a pattern emerged and assuming that you can even, assuming that you don't make any mistakes as well.
00:51
It's very easy to make a mistake in just one step doing all these derivatives and then your whole answer will be off.
00:57
But these particular problems are special.
01:00
And i'm going to show you why we don't actually have to do a lot of work in figuring out what their mclaurin series are, because they're special, and i'll show you.
01:10
So we'll start from, we'll start from easiest to hardest.
01:14
So the easiest one, i think, is c.
01:18
And the thing you have to know to evaluate this problem, c, to do it quickly, you could apply this formula, take all n derivatives and evaluate them at zero and get the mclaren series that way.
01:32
Or you can be smart about it and know that cosine.
01:38
Now, this is a very famous mclaurin series.
01:40
Cosine of x for any x in the real numbers is equal to, it's equal to, oops, from n going from zero to infinity, it's equal to negative 1 to the n, x to the 2n power over 2n factorial.
02:13
And if you didn't know this fact, this is when you should commit to memory, that this is what the mclaurin series for cosine looks like.
02:20
If you've never done so, why don't you use this formula to prove it? because the derivatives of cosine are pretty easy to find.
02:29
You should definitely know what the derivatives of cosine and sine are at this point.
02:34
But anyway, we're going to just use this.
02:36
We're going to take this fact for granted right here because then all we have to do is say that when f of x is equal to x cosine of x then that means that f of x is equal to x times the power series from n going to zero going from zero to infinity negative one to the n x to the 2n over 2n factorial but then we just take this x right here we just take that x and multiply it here and so that's just equal to the power series starting at zero going to infinity of negative one to the n now x to the 2n times x to the 1st is x to the 2n plus 1 all over 2n factorial and so we're done that is the power series that is excuse me the mclaurin series for this function in part c so we've done it it's equal to this series right here.
03:56
Likewise, you should know we're going to do b now.
04:05
And what you should know is that if you know the power series for or the mclaurin series for cosine, you should also commit to memory the mclaren series for sine.
04:19
And it is sine of x is equal to the sum from n equal 0 to infinity of negative 1 to the n.
04:36
That's to say negative 1 to the n plus 1 times x to the 2n plus 1 all over 2n plus 1 factorial.
04:56
Therefore, when our function is sine pi x, all you have to do is to replace x up here.
05:08
All you have to do is to replace x up here this x fella with pi x so you get the series n equals 0 to infinity of negative 1 to the n plus 1 times pi x to the 2n plus 1 all over 2n plus 1 factorial and we're done we found the macloren series for the second guy.
05:46
Now, part a is the hardest.
05:50
And for part a, we're going to use another fact that you should probably know.
05:56
For part a, we're going to use the fact that 1 over 1 minus x is equal to the sum of x to the nth power, where n goes from 0 to infinity, provided the absolute value of x is less than 1.
06:27
So this is a convergent power series, the formula for a convergent power series.
06:35
And we can use it in this case because we can write for our f of x, this is part a, by the way.
06:43
For our f of x, this is equal to 1 over 1 minus x squared squared.
07:00
So i'm going to write that in a slightly weirder way.
07:03
I'm going to write it as 1 over 1 minus x squared times.
07:07
1 over 1 minus x squared.
07:10
Now the thing to notice is that, granted the guy in red that i wrote, this stuff, we can apply that here, and we can say that 1 over 1 minus x squared, it's like saying just swap out this x and this x with an x squared, and we can say that this is equal to sum x squared.
07:44
To the n, where n goes from zero to infinity, which is to say you can just simplify that to n going from zero to infinity of x to the 2n power.
08:04
And provided that the absolute value of x is still less than one, technically we need x squared to be less than one.
08:17
But that's the same thing as saying the absolute value of x less than one.
08:21
So nothing changes there...