Recent content by jgill

Some years back I wrote a BASIC program to search for zeros of complex functions by using a simple "follow the greatest decline" principle (avoiding the calculus) in searching for a minimal of the absolute value of the complex function. I resurrected it and entered your function and got z=10...

Retired (2000) professor of mathematics from a small state university. Minor research background.

Since each term involves the number of terms in your (partial) series I suggest you take a look at Tannery's theorem. Once the N is out of the way, what remains is clearly convergent. What it converges to, however, is another matter. Probably not a closed form.

\mathop L\limits_{k = 1}^n {g_k}(z) = {g_n} \circ {g_{n - 1}} \circ \cdots \circ {g_1}(z) {G_n}(z) = \mathop R\limits_{k = 1}^n {g_k}(z)\mathop L\limits_{k = 1}^\infty {g_k}(z) = \mathop {\lim }\limits_{n \to \infty } {G_n}(z) and \mathop R\limits_{k = 1}^n {f_k}(z) = {f_1} \circ {f_2} \circ...