Using Taylor Series to find Maclaurin Series for exp(x)

In this segment we're asked to derive the Maclaurin series for e^x. So, Maclaurin series is to write down our Taylor series around the point x equal to 0. So, let’s go and first write down what the Taylor series is, the general form of it.

And this is what we have. It has infinite number of terms like that. Then since x is equal to 0, that's what Maclaurin series is all, about no matter what the function is. We'll get something like this that we have to calculate the value of the function and it is derivatives at x equals zero. Now the function which is given to us is e^x, right? We're asked to find out hey what is the, what is the Maclaurin series for e^x. So, if we go ahead and do this so we'll find out that f′(x) will be also e^x, f′′(x) will be also e^x , f′′′(x) will be also e^x, and this can keep on going on as long as we want to. And if we want to find the value of the function at 0, it is 1, because it is just e^0. If we want to find the first derivative of the function at 0, it will be also 1. If we want to find the second derivative of the function at 0, it will be also 1. And if we want to find the third derivative of the function at 0, which is just substituting x equal to 0 here, it'll be also 1. So, we have all the ingredients which are available to us so let's go and see what happens now with these substitutions.

So with these substitutions, we'll get f(h) is equal to f(0), which is 1, plus f prime of 0 which is also 1, times h plus the second derivative of the function at 0, which is also 1, times h squared by 2 factorial, and then the next term is the third derivative of the function at zero, which is also one, and times h cubed by three factorial and so on so forth. So, if we simplify this given, we get this expression right here. But we also recognize that the function which is given to us f of x is e^x, so this is nothing but e^h, f of x is e^x so we'll be f of h e to the power h so that is the Maclaurin series for e to the power h.

Now, people might say hey, I can write in terms of x, yeah since h is a dummy variable in here, I can substitute back, I can substitute for h, I can substitute x, I get 1 plus x plus x squared by factorial 2, plus x cubed by factorial 3, plus so on and so forth. And that is the Maclaurin series for e to the power x. And that's the end of this segment.