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The Troublesome Endpoints of a Trigonometric Series

Date: 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/ 
Associated Topics:
High School Calculus
High School Sequences, Series
High School Trigonometry

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