I came across this series in a book and the author tells the upper bound and it checks out under the conditions but i could n't find the connection or the reason if u may, that had i not been presented with this info, that i could have used to find the upper bound myself.
For $s>1$ the sum of $2^{p}-1$ terms of the series is less than $\dfrac {1} {1^{s-1}}+\dfrac {1} {2^{s-1}}+\dfrac {1} {4^{s-1}}+\dfrac {1} {8^{s-1}}+\ldots +\dfrac {1} {2^{\left( p-1\right) \left( s-1\right) }} < \dfrac {1} {1-2^{1-s}}$
Any hints or clues would be much appreciated.