# Using Taylor's Theorem, find a power series expansion for the following: `e^(x)`

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### 1 Answer

A Taylor Series is one way to represent a function as an infinite sum. The general Taylor Series of a function, f(x) is given by:

`f(x) = f(a) + f'(a)(x-a) +(f''(a)(x-a)^2)/(2!) + ... + (f^(n)(x-a)^n)/(n!) + ...`

Now, we want to find the expansion for `f(x) = e^x.`

Using the expansion series for f(x), we get the following:

`f(x) = e^x = e^a + e^a (x - a) + (e^a (x-a)^2)/(2!) + ... + (e^n(x-a)^n)/(n!) + ...`

or

`f(x) = e^x = e^a [1 + (x-a) + ((x-a)^2)/(2!) + ((x-a)^3)/(3!) + ...]`

A more compact way of writing this is:

`f(x) = e^x =sum_(n=0)^oo (e^a (x-a)^n)/(n!).`

The power series expansion for e^x is the same Taylor Series, at a = 0. Hence, ``

`e^x =sum_(n=0)^oo (x^n)/(n!).`

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