Taylor expansion technique

  1. Hi everyone,
    Is there a certain technique or a program for converting Taylor expansion to summation notation form and vice versa.
    Thank you in advance.
     
  2. jcsd
  3. Stephen Tashi

    Stephen Tashi 4,410
    Science Advisor
    2014 Award

    Your question isn't completely clear. Can you give a specific example where converting from one notation to the other is difficult?
     
  4. let's say I expend a certain function using Taylor series. Is there a specific method I can apply to represent that string of terms in sigma notation.
     
  5. Unless you can find an expression for the nth derivative of the function at a certain point in terms of n, there's no point in trying.
     
  6. Stephen Tashi

    Stephen Tashi 4,410
    Science Advisor
    2014 Award

    Well, that's more specific question, but not a specific example.

    I think your question amounts to asking whether there is a concise way to represent the n-th derivative of a particular function ( like f(x) = (x sin x)/(x+3) ) as an expression with a finite number of symbols in it that only involves specific functions and the variables 'x' and 'n'.

    I don't know of any technique that works for all functions. The higher derivatives of some functions involve more and more terms. You might have to write sums-of-sums or sums-of-sums-of-sums to represent them.

    You could approach the problem as a task in computer algebra. It would involve algorithms that manipulate strings. This makes it a very specialized question. I don't know whether any programmers doing computer algebra hang-out in the computer sections of the forum. I don't recall seeing any computer algebra algorithms discussed in these mathematics sections.
     
  7. Let's say i need to rewrite 2+7(x-2)+4(x-2)^2+(x-2)^3+O((x-2^4) in sigma notation.Is there any systematic way to do that?
     
  8. Mark44

    Staff: Mentor

    To write a series in summation notation, you have to have a general pattern for the n-th term of the series. I don't see any particular pattern in what you showed.
     
  9. Maybe different example: 1 + x + (5/4)x^2 + (7/4)x^3 +...+O(x^4). I am looking for general approach for rewriting expansions like this in sigma notation.
     
  10. There are infinitely many functions f such that f(0)=1*0!, f(1)=1*1!, f(2)=5/4*2!, f(3)=7/4*3!. Without knowing every term, it's impossible to find a summation that continues to be consistent with the taylor expansion of the function forever, in this case, we need the O(x^4)'s expansion.

    (Or you can just use your induction skills to find [itex]f^{\left(n\right)}\left(k\right)[/itex] in terms of n and k to find the expansion around k.)
     
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