Series approximation graphics

Series values along the real axis

Plots of the first terms of the Taylor series of along the real axis. As increases, the curves vary from red to violet. Because the series misses even terms, the curves for the pairs and , and , and coincide.

Plots of the first terms of the Taylor series of along the real axis for . As increases, the curves vary from red to violet. The radius of convergence of the series is π/2.

Series values in the complex plane

Curves connecting points and by summing successively terms of the Taylor series. The number of terms summed increases from for the lower plane to for the upper plane.

Zeros of the series in the complex plane

Roots of the first terms of the Taylor series of in the . As increases from 0 to 50, the roots are colored from red to violet and accumulate on a circle of radius around the origin.

Curves of vanishing real part (red) and imaginary part (blue) of the Taylor series of in the . Terms up to order are taken into account.

Contour plot of the argument of the Padé approximant of the function in the . The coloring is cyclical from red () to red ().