Let $(_)$ $≥0$ be a strictly decreasing sequence of positive real numbers, and let $∈ℂ$, $||<1.$ Prove that the sum $_0+_1 +_2^2+⋯+_^+⋯$ is never equal to zero.
Is it valid to say that since the series converges absolutely, we may take derivatives which then implies it is holomorphic. Holomorphic implies the existence of a convergent Taylor series and by the uniquety of the Taylor series, the polynomial being $0$ implies $a_n$ is $0$, a contradiction?