How to Convert Taylor Expansion to Summation Notation and Vice Versa?

[Vola]

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.

 

[Stephen Tashi]

Is there a certain technique or a program for converting Taylor expansion to summation notation form and vice versa.

Your question isn't completely clear. Can you give a specific example where converting from one notation to the other is difficult?

 

[Vola]

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.

 

[Whovian]

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.

 

[Stephen Tashi]

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.

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.

 

[Vola]

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?

 

[Mark44]

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?

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.

 

[Vola]

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.

 

[Whovian]

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 $f^{\left(n\right)}\left(k\right)$ in terms of n and k to find the expansion around k.)