## Upper bound on $\sum_{k=1}^T \frac{1}{k (1+a)^{T-k}}$

Is there any reasonable upper bound for the following quantity $$\sum_{k=1}^T \frac{1}{k (1+a)^{T-k}}$$

where $$a>0$$ with respect to $$T$$ and $$a$$ (something like $$\mathcal{O}(\frac{\log (T)}{aT}$$)? I tried to compute integral $$\int_{0}^T \frac{1}{x (1+a)^{T-x}}dx,$$ which should be upper bound on this sum as $$f(x) = \frac{1}{x (1+a)^{T-x}}$$ is decreasing on $$(0, T)$$, but I did not achieve to get reasonable expression.