Maclaurin Series for Tangent

Date: 09/17/2001 at 13:46:39
From: Daniel Sutton
Subject: Maclaurin series

Just wondering if you knew the Maclaurin series for tangent (NOT 
inverse tangent).  I've tried doing it manually and on a program 
called MATLAB, but I haven't been able to find a pattern.

Thanks.

Date: 09/17/2001 at 14:47:30
From: Doctor Rob
Subject: Re: Maclaurin series

Thanks for writing to Ask Dr. Math, Daniel!

It is not very surprising that you haven't found a pattern.  The
pattern is very subtle.  The series is

            infinity
   tan(x) =   SUM   2^(2*n)(2^(2*n)-1)*B[n]*x^(2*n-1)/(2*n)!,
              n=1                                         |x| < Pi/2,

                                infinity
   B[n] = 2*(2*n)!/(2*Pi)^(2*n)   SUM   1/k^(2*n)
                                  k=1
        = nth Bernoulli number.

Bernoulli numbers are occasionally encountered in combinatorial
counting problems. They are always rational numbers, and satisfy
certain recursion relations which allow them to be calculated fairly 
economically. The first few are

   B[1] = 1/6
   B[2] = 1/30
   B[3] = 1/42
   B[4] = 1/30
   B[5] = 5/66
   B[6] = 691/2730
   B[7] = 7/6
   B[8] = 3617/510

If you need further assistance, write again.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/