# How to bound $\sum_m e^{\mathcal{O}(k2^{m/2})-2^{3m/4}}$

I have to estimate the following sum $$\sum_{m=0}^{\frac{2}{\log 2}\log\log\log T}e^{\mathcal{O}(k2^{m/2})-2^{3m/4}}.$$ I would like to show that this sum is $$\ll_k 1$$ and if possible that it actually is $$=e^{\mathcal{O}(k^3)}$$ I guess that I can actually sum over all $$m$$ and consider $$\sum_{m=0}^{\infty}e^{\mathcal{O}(k2^{m/2})-2^{3m/4}}$$ and still get the above estimate. I don't know how I should deal with the "big oh" in the exponent while summing. Do you have any hint on how I should proceed? Thank you for your help!