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Topic: Convergence arctan taylor searies
Replies: 1   Last Post: Feb 20, 2013 7:35 PM

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wilson

Posts: 4
Registered: 9/27/12
Re: Convergence arctan taylor searies
Posted: Feb 20, 2013 7:35 PM
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On Wed, 20 Feb 2013 19:04:46 -0500, Dave Rudolf <dave.rudolf@usask.ca>
wrote:

> Hey folks,
>
> I was looking at the taylor series that is commonly used to compute
> arc-tangent, which is
>
> atan(x) = x - x^3/3 + x^5/5 - x^7/7 + x^9/9....
>
> However, the domain for arc-tangent is +/- infinity. So, for x values
> that are a bit larger than 1 (or smaller than -1), I don't see how
> this series converges to anything. For instance, if I did atan(2),
> then (2)^n will grow a lot faster than n itself, so each term will be
> larger than the last.
>
> Or am I missing something here?
>
> Thanks.
>
> Dave



Yes Dave, you are right. The above series converges only for x^2 <= 1.

"Table of Integrals and Products" by Gradshteyn and Ryzhik in Eq. 1.644
gives a series expansion that holds for all values of x^2 < infinity.
(The book should be available in any good college library.) It is much
too messy to try and reproduce here. And probably not a good way to
compute the atan.

Keep uup the good work.

Lee

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