Let $n_i$, $i=1,\ldots,m+1$ be nonnegative natural numbers, sum of which $\sum_{i=1}^{m+1}n_i=N$.
I woul like to find an upper bound for the following$ \sum_{i=1}^{m+1}\frac{\sqrt n_i}{2^{i-1}}$
Let $n_i$, $i=1,\ldots,m+1$ be nonnegative natural numbers, sum of which $\sum_{i=1}^{m+1}n_i=N$.
I woul like to find an upper bound for the following$ \sum_{i=1}^{m+1}\frac{\sqrt n_i}{2^{i-1}}$
Using $x=(\sqrt{n_1},...,\sqrt{n_{m+1}})$, $y=(1,\frac{1}{2},...,\frac{1}{2^m})$, the Cauchy-Schwarz inequality (