Maclaurin Series for e^x

Deriving an expansion series for e^x aka (e to the power x) is very easy to do. This is the easiest one of all because the derivative of e^x is e^x. It does not change no matter how many times you differentiate it.

Here we have f(x) = e^x, and when x = 0 the result is 1 because anything to the power 0 is 1 in algebra.

Differentiating e^x gives us e^x again, and when x = 0, the result is 1.

Differentiating e^x gives us e^x. Are you seeing a pattern?

As you can see, we always get the same result.

All the terms will have +1in them.

The pattern for this function is therefore, +1, +1, +1, +1...

As you can see, I have substituted the results into the Maclaurin general formula to give what you see above. It is just a simple power series!

That is amazing, who would have ever thought that e^x could be written as an expansion series.