# Taylor series for tanh

1. Mar 19, 2009

### kercalc

Hi.

How can I derive the Taylor series expansion and the radius of convergence for hyperbolic tangent tanh(x) around the point x=0.

I can find the expression for the above in various sites, but the proof is'nt discussed. I guess the above question reduces to how can I get the expression for the n^th order derivative of tanh(x) in terms of Bernoulli numbers.

Many thanks in advance.
joel.

2. Mar 19, 2009

### mathman

The radius of convergence for tanh is the same as it is for tan, which is pi/2.

3. Mar 20, 2009

### kercalc

Thanks. But my questions are (a) How can I derive (not just the expression) the n^{th} coefficient of the Taylor series for tanh(x) around x=0, and (b) the derivation of the radius of convergence (by e.g. root test).