Complex exponential series

The elements of the complex exponential sequence follow the rule

zn+1= zn*a / n

(for comparison:  geometric series :  zn+1= zn*a )

zn i the nth member of the sequence, with index n a positive entity, including 0. The growth parameter a is a complex number. With

z0= 1

the members are: 1, a /1 , a 2/(1*2), a3/(1*2*3), a4/(1*2*3*4)....

zn =an/ n!

( n! = 1*2*3*4*...*n n! = n -faculty)

The complex exponential partial sum series Sn is formed by consecutive addition of the members of the sequence:

Sn= Σ0n am/m! 0 ≤ m ≤n ; Sn = 1 + a + a 2/2....+a n/n!

This simulation calculates 500 elements of the sequence. By drawing the red point in the left chart, a is defined. The blue points are the members of the sequence. In the right chart they are those of the partial sum series.

The case of the real exponential series is observed when a is a point on the real axis.

Members of the exponential sequence always converge to zero. Its partial sum series converges to a finite number for all finite a.

lim (Sn= Σ0n am/m!) = ea; e = 2.71828....Euler number

When a has an imaginary component the series spirals toward the convergence point (limit). Its sourrounding is marked by a small green circle.

For small imaginary parts all points will be on a single Riemann sheet. For large imaginary parts of a one observes multiple revolutions of the spiral, corresponding to points in different Riemann sheets.  The effect is less obvious than with the complex geometric series because of the fast convergence of the exponential, for which most calculated points lie within the small green circle.