experimenting-with-a-sequence-of-complex-numbers

Experimenting with a sequence of complex numbers

Consider the sequence S(n, z) = (1 - z)(z + z**2 + z**3 + ... + z**n) where z is a complex number and n a positive integer (n > 0).

When n goes to infinity and z has a correct value (ie z is in its domain of convergence D), S(n, z) goes to a finite limit lim depending on z.

Experiment with S(n, z) to guess the domain of convergence Dof S and lim value when z is in D.

Then determine the smallest integer n such that abs(S(n, z) - lim) < eps where eps is a given small real number and abs(Z) is the modulus or norm of the complex number Z.

Call f the function f(z, eps) which returns n. If z is such that S(n, z) has no finite limit (when z is outside of D) f will return -1.

Example

I is a complex number such as I * I = -1 (sometimes written i or j).

f(0.3 + 0.5 * I, 1e-4) returns 17

f(30 + 5 * I, 1e-4) returns -1

Remark

For languages that don't have complex numbers or "easy" complex numbers, a complex number z is represented by two real numbers x (real part) and y (imaginary part).

f(0.3, 0.5, 1e-4) returns 17

f(30, 5, 1e-4) returns -1

Note

You pass the tests if abs(actual - exoected) <= 1