An Elementary Difference–Matrix Evaluation of the Series An Elementary Difference–Matrix Evaluation of the Series ∑(n=1 to ∞) nᵏ/2ⁿ

We present a fully elementary method for evaluating the infinite series $S_k = \sum_{n=1}^{\infty} \frac{n^k}{2^n}$, where k is a fixed natural number. The method relies only on repeated scaling, term-by-term subtraction, and the systematic use of finite differences. No tools from calculus, generating functions, or special functions are required. Starting from explicit computations for $k=1,2,3,4$, we show how a stable pattern emerges and how this pattern can be described and proved using a difference matrix. Finally, we present an interesting combinatorial identity.