MTH 251 Lecture Notes - Lecture 31: Convergent Series

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Section Taylor and McLaren series
Two big question:
What functions have power series representations and how can we find them?
Suppose has a power series of a with radius ; that is,

If
Differentiating, we get:

If
Repeating: 

If  .
 Derivative

If 

If 
In general, it seems that
Theorem:
If has a power series expansion (or representation at that is, if

then the coefficients are given by,


Replacing in the formula, we have
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