Convergent Power Series of sech(x)‎ and Solutions to Nonlinear Differential Equations

Abstract EN

It is known that power series expansion of certain functions such as sech(x) diverges beyond a finite radius of convergence.

We present here an iterative power series expansion (IPS) to obtain a power series representation of sech(x) that is convergent for all x.

The convergent series is a sum of the Taylor series of sech(x) and a complementary series that cancels the divergence of the Taylor series for x≥π/2.

The method is general and can be applied to other functions known to have finite radius of convergence, such as 1/(1+x2).

A straightforward application of this method is to solve analytically nonlinear differential equations, which we also illustrate here.

The method provides also a robust and very efficient numerical algorithm for solving nonlinear differential equations numerically.

A detailed comparison with the fourth-order Runge-Kutta method and extensive analysis of the behavior of the error and CPU time are performed.