Power Series

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Power Series
Power Series
How does the taylor power series for log(1+x)/x?


I know the power series for log(1+x) = x - (x^2)/2 + (x^3)/3 - ...,

Here it is:

http://www26.wolframalpha.com/input/?i=taylor+series+for+log(1%2Bx)%2Fx&asynchronous=false&equal=Submit

It's the series you wrote for log(1+x) with every term divided by x reducing its power by one.



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Power Series Representation of Functions

Is there a general closed form for the reciprocal of a power series or a Taylor series?


So, if I have a power series a_0+a1*x+a_2*x2+...+a_n*x^n+... , how do I find another power series b_0+b_1*x+b_2*x2+...+b_n*x^n+... such that the two power series multiply to 1. I know how to get the recursive expression for b_n, but I'm wondering whether a nonrecursive one exists. This will let me divide Taylor polynomials, to get expressions for tan(x), which I've hurt to be difficult, leading me to believe that there is no closed form for the reciprocal of a power series.

I'm not sure what you mean by "I know how to get the recursive expression for b_n". I've never seen recursion used in a Taylor series.

For any function f(x) for which you have a series, you can just as well do a series expansion on 1/f(x) which is the reciprocal series.

It sounds like you are trying to expand tan(x) so you want a reciprocal series to cos(x) to do it. The tan(x) expansion isn't that bad since d/dx (tan x) = 1 + tan(x)^2. If you do the expansion on 1/cos(x) you will find it looks just like the expansion of tan(x) because d/dx (1/cos x) = tan(x)/cos(x) so the derivatives are just like the tangent derivatives.

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