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So I managed to show that the Taylor series of any function expressible as a power series is as such : https://www.dropbox.com/s/pkl3krcwafgmsx2/Taylor%20Series.pdf?dl=0

but I have no idea how to show that any infinitely differentiable function can be written in the form of a power series. Why is the power series such an all encompassing series? I asked a friend and he brought up something about it being constructed abstractly with sets which didn't make any sense to me.

Chung Ren Khoo
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    Not every function can be written as a power series. – Angina Seng Dec 20 '17 at 11:25
  • sorry, I meant to say any infinitely differentiable function. – Chung Ren Khoo Dec 20 '17 at 11:31
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    Not every infinitely differentiable function can be written as a power series. – Angina Seng Dec 20 '17 at 11:32
  • so what type of functions can be written as a power series? or would it be easier if I asked, what type of functions cannot be written as a power series? – Chung Ren Khoo Dec 20 '17 at 11:33
  • I could say that the functions that can be locally written as power series are the real analytic functions, but that is tautologous, as by definition, real-analytic functions are those whose Taylor series at each point converge to the function in a neighbourhood of that point. – Angina Seng Dec 20 '17 at 11:40

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For real functions it is simply not true that every infinitely differentiable function can be written as a power series. The common counterexample to this is $$ f(x) = \begin{cases} 0 & x=0 \\ e^{-1/x^2} & x\ne 0 \end{cases} $$ which is infinitely differentiable everywhere (it takes a bit of work to show this at $0$), but all of its derivatives at $0$ are $0$, so its Taylor series is $\sum_n 0\cdot x^n$ which does not converge to $f(x)$ anywhere except at $x=0$.

For complex functions things are much nicer: If a function is even differentiable once everywhere on a disk, it can be written as a power series around the center of that disk. But proving that is much more involved than what you link to.

A real function that can be written as a power series in some neighborhood of every point is called real analytic. There is not really any nice characterization of this other than "has power series everywhere", as long as if we're only looking at real numbers. If we involve complex numbers, the real analytic functions are exactly those functions that can be written as a restriction of a complex function that is differentiable on an open set that includes the real line.

hmakholm left over Monica
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  • I've never had any formal education in mathematics so I'm a bit lost. Would it be correct if I say : For any infinitely differentiable function that can be written as a power series, the Taylor series expansion exists for that function? And would the converse be true : For a function that cannot be written as a power series, the Taylor series expansion of said function does not exist? – Chung Ren Khoo Dec 20 '17 at 12:04
  • @ChungRenKhoo: Yes: if a function can be written as a power series at all (around some point), then that power series is exactly its Taylor series (around that point). – hmakholm left over Monica Dec 20 '17 at 12:14
  • Thank you. That is exactly what I was hoping to hear. Now it makes a lot more sense. – Chung Ren Khoo Dec 20 '17 at 12:15
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I guess the point your friend brought up must be the following: Historically, function meant power series. Let me explain. Historically, only functions that can be written as a power series were deemed functions. This directly comes from practice of course.

The abstract notion of function as a pair of sets and a subset of their cartesian product must have appeared in 19th century, and I am guessing is due to Cantor.

ugur efem
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  • Huh. That would explain his bizarre answer. His exact words were "It's constructed abstractly and well embedded in analysis" though at that point I had no idea what to make of it. – Chung Ren Khoo Dec 20 '17 at 12:01