Maclaurin Series for TangentDate: 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/ |
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