Simple Series to calculate tan(x)

[Namir]

The series that I mentioned in the opening of this thread is simpler than the common summation series seen in many websites (which involves using Bernoulli numbers). Using the CORDIC algorithm to calculate the trigonometric functions remains the most efficient way. My main point is to single out a simpler (but not necessarily very efficient) series to calculate tan(x). Willy Kunz's result of 5000 iterations demonstrates the inefficiency of the algorithm. Using today's calculators and computers with their fast CPU can make using the series acceptable. However, the purist mathematician/programmer can still argue that the series is not the best way to calculate tan(x). The same can be said for infinite products that calculates sin(x) and cos(x). Their formulations are simple, but they require a high number of iterations.

I found the series for tan(x) on the bottom of page 83 in the book "Handbook of Computing Elementary Functions", 1965, by Lyusternic et al. The book has a lot of series, infinite products, and various approximations for common functions found in calculators. The original authors are Russian.