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Taylor expansion technique 
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#1
Jan1613, 01:30 PM

P: 14

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
Jan1613, 02:38 PM

Sci Advisor
P: 3,252




#3
Jan1613, 03:01 PM

P: 14

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.



#4
Jan1613, 03:19 PM

P: 642

Taylor expansion technique
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.



#5
Jan1613, 03:29 PM

Sci Advisor
P: 3,252

I think your question amounts to asking whether there is a concise way to represent the nth 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 sumsofsums or sumsofsumsofsums 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 hangout in the computer sections of the forum. I don't recall seeing any computer algebra algorithms discussed in these mathematics sections. 


#6
Jan1613, 03:35 PM

P: 14

Let's say i need to rewrite 2+7(x2)+4(x2)^2+(x2)^3+O((x2^4) in sigma notation.Is there any systematic way to do that?



#7
Jan1613, 03:40 PM

Mentor
P: 21,215




#8
Jan1613, 03:51 PM

P: 14

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.



#9
Jan1613, 03:59 PM

P: 642

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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