Taylor series representation function?

Question

Is there a built in Taylor series representation function in Wolfram Mathematica? For Example:

In=TaylorSeries[(*function goes here*)]

Out=(*Function rewritten in sigma notation as a power series*)

If not, is it possible to code such a thing...? How does it work in Wolfram Alpha?

Yes, but it doesn't wrap up the expression into a generalized sigma notation. It shows the expansion. — useranonis, Jul 28 '16 at 03:40
So basically you need something that can discern the general form of the n'th derivative of a function in an automated way? This seems really difficult, and in general it seems possibly mainly only with a gigantic look-up table. — march, Jul 28 '16 at 03:46
I'm still curious how Wolfram Alpha computes all those different series. Specifically the first that appears, it appears as the most compact. — useranonis, Jul 28 '16 at 04:15
I found an even better way to do it... Please hold on, and keep this thread active. — useranonis, Jul 28 '16 at 05:40

useranonis · Answer 1 · 2016-07-28T06:57:38.437

An example for sine as a function of x:

SeriesRepresentationSin = TraditionalForm[Inactive[Sum][SeriesCoefficient[Sin[x], {x, 0, k}, Assumptions -> k >= 0]*x^k, {k, 0, \[Infinity]}]] (*Outputs function expression as series*)
Activate[SeriesRepresentationSin] (*Outputs algebraic function expression*)

I'm still curious of how Wolfram Alpha outputs so many different series. I'm guessing for now it has to do with the "Assumptions" command.

There is also:

MathematicalFunctionData["Sin", "SeriesRepresentations", "TraditionalFormPresentation" -> True]

Although it appears to be limited to some known functions, I believe.

Taylor series representation function?

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