Taylor-Series Cookbook |
There are many ways of coming up with the Taylor Series of a given function f(x). Here are some of the possible recipes:
The Recipes:
From scratch: Given the formula for f(x), directly compute a_{n} = (f^{(n)}(c))/(n!) to obtain the n_{th} Taylor series coefficient a_{n} for f(x) at c. Then the Taylor series for f(x) is:
a_{n} (x-c)^{n} | |
n=0 |
(ii) Substitution: Write your function f(x) as g(2x), or as g(x^{2}), or as g(h(x)) for some other simple function h(x), where g(x) is a function whose Taylor series you already know, and get the Taylor series for f(x) by plugging in u=h(x) into the Taylor series for g(u).
(iii) Differentiation or Integration: If f(x) = d/dx(g(x)), that is, if the given function f(x) is the derivative of some function g(x) whose Taylor series you know, then the series for f(x) can be obtained from differentiating term-by-term the series for g(x). If f(x) is an antiderivative of a function h(x) whose series you know, you can integrate the series for h(x) term-by-term; however, remember to also solve for the constant of integration "C" in this case.
(iv) Creative Combos: If the Taylor series for f(x) and g(x) are known, with n_{th} terms a_{n} and b_{n} respectively, what is the Taylor series for h(x) if h(x)=f(x)+g(x)? If h(x) = x f(x)?
Note: in these recipes, all Taylor series are centered at the same point, c.
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Cook up a storm by finding the Taylor series for the following cornucopia of functions, as well as determining the radii and intervals of convergence of each Taylor series you find.
1/(x+1) | ln(x+1) | 1/(x+1)^{2} | x/(1-x) |
1/[ 1 + 4x^{2} ] | tan^{-1}(2x) | 1/(1 + x^{2}) | 1/(4 + x^{2}) |
For an extra challenge, try f(x) = ln [(1+x)/(1-x)] -- try rewriting it first -- and f(x) = 2/(3x+4).
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1996, Harel
Barzilai.
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