The Troublesome Endpoints of a Trigonometric SeriesDate: 04/10/2011 at 18:11:43 From: John Subject: Proof for pi series 1 -1/3 + 1/5 - 1/7 + ... Hi, I have a question about the following proof. Start with this series: 1/(1 + x^2) = 1 - x^2 + x^4 - x^6 + ... Integrate both sides: Arctan x = x - x^3/3 + x^5/5 - x^7/7 + ... Let x = 1, then: pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... In the first step, I know that the series expansion for 1/(1 + x^2) is only valid for |x| < 1. So if you integrate it, it is still valid only for |x| < 1, isn't it? But then why would it make sense to let x = 1 in the last step? Thank you. Date: 04/10/2011 at 22:46:37 From: Doctor Vogler Subject: Re: Proof for pi series 1 -1/3 + 1/5 - 1/7 + ... Hi John, Thanks for writing to Dr. Math. That is actually a very good question, and just by asking it you are showing a great deal of mathematical intuition. You are right to question the validity of that proof. In fact, it is not quite complete, for precisely the reason that you asked about. It is true that the first sequence only converges for |x| < 1. The series that comes out of the integral also has the same radius of convergence (which you can easily check for this particular series, but I believe that it is a theorem that term-wise integration or differentiation of a power series results in a power series with the same radius of convergence). But the new series also converges at the two endpoints -- and funny things often happen at the endpoints. Does that affect the validity of the equation? Is the new power series still equal to the arctan of x at the endpoints? The argument really only shows that they should be equal when |x| < 1. It turns out that finishing off the proof requires citing an important but non-trivial theorem attributed to Abel: http://en.wikipedia.org/wiki/Abel%27s_theorem Abel's Theorem says that a power series that converges at one or both of its endpoints actually converges to a continuous function on its entire interval of convergence. So the series is equal on its endpoints to the same function it equals inside the radius of convergence. Then you just need to use the Alternating Series Test to show that your series is convergent at the endpoints. In fact, you'll notice that the series you asked about is one of the series mentioned in the Wikipedia article on Abel's Theorem. If you have any questions about this or need more help, please write back and show me what you have been able to do, and I will try to offer further suggestions. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/ |
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