Abstract
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.
Keywords
Series, Difference Matrix, Combinatorial Identity
How to Cite
Imaninezhad, M., (2026) “An Elementary Difference–Matrix Evaluation of the Series An Elementary Difference–Matrix Evaluation of the Series ∑(n=1 to ∞) nᵏ/2ⁿ”, Ohio Journal of School Mathematics 102(1): 6, 75-80. doi: https://doi.org/10.18061/ojsm.7047