Complex Exponentiation using Infinite Summation

Question

I am trying to create an equation that raises a number (x) to a complex number (a+bi) using only infinite summation. Am I allowed to have nested infinite series inside of infinite series?
$$x^{a+bi}=x^a\bigg[\sum_{n=0}^\infty[\frac{(-1)^n(b\sum_{n=1}^\infty[\frac{{(-1)^n(x^{-1}-1)^n}}{n}])^{2n}}{(2n)!}]+i\sum_{n=0}^\infty[\frac{(-1)^n(b\sum_{n=1}^\infty[\frac{{(-1)^n(x^{-1}-1)^n}}{n}])^{2n+1}}{(2n+1)!}]\bigg]$$

Answer 1

In principle you need to worry about uniform convergence when you do this. For the exponential and the trig functions the denominators increase rapidly enough that you have good convergence, so expressing $e^{a+bi}$ as a nested sum of power series is fine.

But in your expression, you also use the trick of summing a geometric series, and there you have a radius of convergence headache.

Also, one notational quibble: When you express a double sum, please use a different letter for the summation variable of the inner and outer sums.