Okay in this video we're gonna talk about Taylor and Maclaurin Series
And this is stuff that you can definitely spend a while talking about so I'm gonna try and give you the
brief 10 minute synopsis here
The idea is that in certain cases you can't do this all the time
The idea is that you can rewrite a function f(x) in terms of an infinite sum of other functions
And basically the formula that you need to know generically is this
The idea is you're gonna centre this, what's called some value "a"
So it says we write f(a) plus f'(a) times (x-a) to the first over one factorial plus f''(a) times (x-a) to the power of 2
over 2 factorial. This pattern keeps continuing, the derivatives get higher, the powers get higher
the factorials get higher, and this keeps just going on and on forever
and just more compactly using summation notation it says we can write that as the infinite sum
of the "nth" derivative evaluated at a, (x-a) to the nth power over n factorial
The only thing you really have to fill in on these formulas is you have to calculate f(a), f'(a)
you'll drop in your given "a" value, f''(a), again you'll drop in your given "a" value
so on an so forth. So mechanically, they actually turn out to be not so bad to calculate
Y'know, taking derivatives can be tedious. Sometimes they want you to rewrite it in the summation notation
that can also be tedious, but the basic idea isn't really too bad at all
It's one of these things that seems like... It feels worse than it actually is
So let's do a problem
So we're gonna find what's called a Maclaurin Series for f(x) = e^5x
A Maclaurin Series means that you're centring this at the value of a = 0
In the other problem it said we basically needed to start calculating derivatives and plugging in this value so
I'm just gonna make a little table of derivatives
The first derivative I'll get e^5x times 5 or 5e^5x
The second derivative I'm gonna get, well, I'll get 5e^5x times 5. You could write that as 25, one thing
I always recommend is not to multiply things out 'cause you'll see the pattern better that way so I
can write this as 5 squared, e to the 5x
The third derivative I'm gonna get 5 cubed, e to the 5x. And then general my nth derivative, after I've done
some arbitrary number, I'll have 5 to the n times e to the 5x
Now all I'm gonna do is just fill in this formula with given values
So it says my f(a), in this case, will be f(0). If I plug f(0) into this, I'll get e to the zero, which is 1
f'(0), I'll get 5 times e to the 0, or just 5. f''(0) I'm gonna get 5 squared, f'''(0) I'll get 5 cubed
Eventually my nth derivative evaluated at 0, I'll have 5 raised to the n power
Now if you compare this to the formula that I just had, it says we can actually write
e to the 5x, well maybe we can, we're not gonna talk about the justification for when you can or when you can't in this video
It says you'll get f(0), which is 1, plus f'(0), which is 5, then it says we take x minus our "a" value over 1 factorial
Then it says we take f''(0), which will be 5 squared, times (x-0) squared over 2 factorial
and we'll just keep this pattern up indefinitely
You could actually rewrite this a little more compactly. It looks to me like the infinite sum from n=0 to infinity
We'll have the nth derivative, which is gonna be 5^n. And then it simply looks like we have x^n over n factorial
Notice if you plug in n=0 you'll get 5 to the 0, 0 factorial is defined to be 1, and so we will get 1
If you plug in n=1, we'll get our first term, if you plug in n=2 you'll get our second term, if you plug
in n=3, you'll get our third term, so on and so forth
Now this is all considered a Maclaurin series expansion for the function e^5x
This is usually what they'll wanna see, a kind of more compact way of writing it
And then the problem could say, find the radius of convergence or the interval of convergence
and we would simply do the ratio test on this, so if you're not sure about that, I talk about the ratio test and finding
intervals of convergence rather for power series, and you can do all that stuff on this one
I'm already at 6 minutes here and I know there's this 10 minute time limit
I'm gonna easily run over so, I think I'm just gonna make another video and we'll pick up finding the Taylor Series expansion for ln(x) at a=2
If you wanna see another example, dig around, it should be close by