For the following equation, is there a better bound than the trivial bound I proposed $$ \Bigl( a \sum_{x=1}^n [\ x^d-(x-1)^d]\ \exp\Bigl\{-\frac{x^2}{\sigma^2}\Bigr\} +\sum_{x=1}^n x^{d-2}\exp\Bigl\{-\frac{x^2}{\sigma^2}\Bigr\} \Bigr) \\\leq (a+1)\sum_{x=1}^n x^d\exp\Bigl\{-\frac{x^2}{\sigma^2}\Bigr\} ~\text{as}~n~ \text{is large.} $$
This trivial upperbound seems large and I am looking for a tighter one. I thought about $ [\ x^d-(x-1)^d]\ = [\ (x-1+1)^d-(x-1)^d]$ but I don't have an idea how to continue!
I would be thankful for any help
Assuming $d\geq 1$, one has $$ x^d-(x-1)^d=x^d(1-(1-1/x)^d)\leq x^d(1-(1-d/x))=dx^{d-1} $$ using Bernoulli's inequality, which obtains a slightly better bound of $$ (ad+1)\sum_{x=1}^n x^{d-1}\exp(-x^2/\sigma^2). $$