Is there any easy way to determine taylor series for functions?

Question

for functions like f(x) = 1/(1+x^2) or f(x) = atan(x)
I found the series by using a derivative calculator to get f' , f'' . f''' and so on until i could figure out what the series was. but its a pain finding the 3rd or 4th derivative of 1/(1+x^2)

is there a more convenient hands on or systematic way of determining a Taylor series for a function?

alwbsok · Accepted Answer

In certain cases, there is. You can use substitutions and calculus to transform known taylor series into taylor series that you want. For example, I can find the taylor series centred at 0 for 1 / (1 + x^2) by considering the geometric series:

sum n = 0 to infinity of x^n = 1 / (1 - x) ... for |x| < 1

If we replace x with -x^2, we get:

sum n = 0 to infinity of (-x^2)^n = 1 / (1 + x^2) ... for |-x^2| < 1
= sum n = 0 to infinity of (-1)^n x^(2n) = 1 / (1 + x^2) ... for |x| < 1

To get atan(x), we integrate the power series of 1 / (1 + x^2). Integration of power series happens term-by-term, so:

sum n = 0 to infinity of (-1)^n x^(2n + 1) / (2n + 1) = atan(x) + C

When x = 0, left side is - and the right side is C, so C = 0. Therefore:

sum n = 0 to infinity of (-1)^n x^(2n + 1) / (2n + 1) = atan(x)

Hope that helps!

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Is there any easy way to determine taylor series for functions?

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