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

Abstract

It is known that power series expansion of certain functions such as $\mathrm{sech}(x)$ diverges beyond a finite radius of convergence. We present here an iterative power series expansion (IPS) to obtain a power series representation of $\mathrm{sech}(x)$ that is convergent for all $x$. The convergent series is a sum of the Taylor series of $\mathrm{sech}(x)$ and a complementary series that cancels the divergence of the Taylor series for $x\ge \pi /\mathrm{2}$. The method is general and can be applied to other functions known to have finite radius of convergence, such as $\mathrm{1}/(\mathrm{1}+x^{\mathrm{2}})$. 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.