8.4. Taylor Series

Example 8.4.2 (a): Derivatives of Power Series

Assume that f(x) = e2x has a convergent power series expression centered at c = 0, i.e. e2x = anxn. Find the coefficients an.

According to our theory we know that if our function can be represented as

e2x = anxn = = a0 + a1x + a2x2 + a3x3 + ...


an =

Thus, to find the coefficients an we bascially need to find the n-th derivative of the function f at the center of the series. Taking derivatives on both sides and substituting 0 (the center in this example) we get:

n=0 f(x)=e2x f(0)=1
n=1 f '(x)=2 e2x f '(0)=2
n=2 f ''(x)=22e2x f ''(0)=22
n=3 f (3)(x)=23e2x f (3)(0)=23

and so on, so that by "poor man's induction" we get:

an = 2n/n!

Therefore, if e2x had a series representation, it would have to be:

e2x = 2n/n! xn
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