Upper estimate of a combinatorial sum

Abstract

Abstract We give an upper estimate for the binomial sum U d x = ∑ r ≥ 0 d − r + 1 r ⋅ x r $\begin{array}{} U_{d} \left(x\right)=\sum _{r\ge 0}\binom {d-r+1} r\cdot x^{r} \end{array} $ with natural d and real nonnegative x. In particular, this estimate implies that U d x = O ( ( 0.5 + x + 0.25 ) d ) $\begin{array}{} U_{d} \left(x\right)=O\bigl((0.5+\sqrt{x+0.25} )^{d} \bigr) \end{array} $ with fixed x > 0 and d → ∞.