MathCS.org - Real Analysis: Example 8.4.2 (a): Derivatives of Power Series

Example 8.4.2 (a): Derivatives of Power Series

Assume that f(x) = e^2x has a convergent power series expression centered at c = 0, i.e. e^2x = a_nxⁿ. Find the coefficients a_n.

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

e^2x = a_nxⁿ = = a₀ + a₁x + a₂x² + a₃x³ + ...

then

a_n =

Thus, to find the coefficients a_n 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)=e^2x f(0)=1

n=1 f '(x)=2 e^2x f '(0)=2

n=2 f ''(x)=2²e^2x f ''(0)=2²

n=3 f ⁽³⁾(x)=2³e^2x f ⁽³⁾(0)=2³

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

a_n = ^2ⁿ/_n!

Therefore, if e^2x had a series representation, it would have to be:

e^2x = ^2ⁿ/_n! xⁿ

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

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Example 8.4.2 (a): Derivatives of Power Series