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!


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