2

I am looking for syntax for WolframAlpha, which would evaluate a series at a given point. My goal is to have a sheet of practice exercises with links to solutions in WolframAlpha.

Example problem: Approximate the value of $\sqrt{102}$ using Taylor series of appropriate function at appropriate point of 2nd order.

The solution on paper is straightforward: calculate a Taylor series of 2nd order of $f(x)=\sqrt{x}$ at $x_0=100$ and plug in $x=102$ . In WolframAlpha, I can get the Taylor series with a query such as "Taylor series of sqrt(x) at x=100 to order 2". I can also evaluate any expression via "evaluate x^2+2x+1 at x=102". Is there any way to evaluate the series at a given point?

I tried many combinations of verbose commands as well as direct Mathematica inputs, but nothing seems to work. I have a feeling it has something to do with the error term.

Lukáš Mrazík
  • 97
  • 1
    The Mathematica command that will get you this is Normal[Series[Sqrt[x], {x, 100, 2}]] /. {x -> 102} however I'm not sure you can get Wolfram Alpha to parse it as such. – user3257842 Mar 22 '23 at 10:05

1 Answers1

1

This is quite a hacky solution, but you can nest "Limit", Sum, and SeriesCoefficient to produce what you want, although WA is very particular about the syntax (link):

Sum[SeriesCoefficient[Sqrt[100+x], {x, 0, n}] * x^n, {n, 0, 2}] as x->2.0
                                                            ↑
                                                          degree

Neither using the symbol Limit nor natural language, e.g. Limit[Sum[...], x->2.0] / limit Sum[...] as x->2.0, returned any useful results. Replace 2.0 with 2 if an exact result is desired.

I'm including a screenshot of the output for posterity:

enter image description here

user170231
  • 19,334
  • Thank you, I never expected such an old question to be answered. I am actually impressed that this input also returns the Taylor series (labeled as such). Also, I was unaware of the 2.0 vs 2 trick, I shall use it in the future. – Lukáš Mrazík Feb 27 '24 at 08:58
  • You're welcome! Though just to be clear, the displayed "Series expansion at $x=2$" is literally the series expansion of the limand at $x=2$, i.e. the Taylor series of the degree-$2$ Taylor series. It's not particularly useful in its given form, but it is identical to the $2^{\rm nd}$-order approximation that the prompt is having us use. – user170231 Feb 27 '24 at 16:56