Bounding a summation of positive terms

Bound the summation of the form

$ \sum_{k=\tau}^{t} (1+k)^{-v} $

where $ \tau $ and t are positive integers and v $ \in $ (0.5, 1].

This summation is bounded by the form $ (1 + t - \tau)^{-v} / (1 -v) $

I am not able to obtain this form, while bounding using integral I am getting $\frac{(1 + t)^{1 -v} - (1 + \tau)^{1-v}}{1 -v}.$

I am not able to figure out how to obtain the bound give before using the latter (or whether this can be used to achieve that).

PS: This is in one of the research paper that I am currently studying. Research paper page number 26