# Power series approximation for any function such as $-e^{x^{2}}$ in some easily-accessed open-source software?

My comrades repeatedly encourages monotonous problems where the issue is the same: chain-rule and some basic arithmetic. Is there some computational way to derive power series approximations? Suppose I plug in some function such as $e^{x^{2}+x+ln(x)+x^{777}}$, $e^{x^2}$ or $x^{777}-ln(x)+e^{2x+1}$, some command such as Series(e^{x^{2}}) to get you the series? It would be useful to see graphically how things change, which would help me to learn things graphically.

• FWIW if open-sourceness is not a requirement, I think Mathematica (or Wolfram Alpha) is currently the most user-friendly tool for this sort of thing. – David Z Feb 5 '12 at 6:28
• Isn't this just your question Is there any open-source or easy-to-access software that can... with a different example problem? – Mark Booth Feb 6 '12 at 14:09

You have a few options:

• Sage: Documentation for power series is here. (Scroll down until you reach "Power series.") Documentation for plotting functions is here.

sage: var('x')
x
sage: f = exp(x^2 + x + log(x) + x^777)
sage: f.taylor(x, 0, 3)
3/2*x^3 + x^2 + x

• SymPy (Right on that page, first heading) + Matplotlib

• Maxima (Search for "Taylor series" on that page.) + Gnuplot (or Matplotlib, or another plotting package you prefer)

For any of the software packages listed in your earlier question on open-source math software packages, you can probably search around and find the appropriate command.

• SymPy, Matplotlib and Maxima are all built into Sage, so you only need to install one package if you want to try them all out. – Dan Feb 4 '12 at 8:24