This document discusses unbounded transcendent formal power series over a finite field Fq((X-1)). It presents a new general result that establishes a transcendence criterion for continued fractions with unbounded partial quotients constructed from algebraic elements. Specifically, the theorem shows that if a formal power series can be approximated by a family of algebraic series with increasing block lengths, then the formal power series is transcendental. The proof uses previous results on continued fractions over finite fields and algebraic degree estimates. An example is also given to illustrate the main result.