sum of square roots upper bound

Question

I want to know whether there exists a constant $c$ such that for any $k$ the following inequality holds: $$\sum_{i=1}^{k}\sqrt{2^i} \leq c \sqrt{2^k}$$

This is not a homework. Could somebody provide me with a proof or at least a pointer on how to solve this?

Answer 1

Note that $$ \sum_{i=1}^k \sqrt{2^i} = \sum_{i=1}^k \left(\sqrt2\right)^i = \frac{\left(\sqrt2\right)^{k+1} - 1}{\sqrt2 - 1} \le \left(\sqrt2\right)^k \frac{\sqrt2}{\sqrt2 - 1} $$

So let $c$ be any number at least $\frac{\sqrt2}{\sqrt2-1} \le 3.415$

Answer 2

You can turn the sum into an integral by considering it a Riemann sum of the function $2^{x/2}$. You can bound the sum between the integral from $0$ to $k$ and the integral from $1$ to $k+1$