What can be said about this sequence:
$\displaystyle 1 + \surd2 + \surd3 + 2 + \surd5 + \surd6 + ..... + \surd n $
Hint: it might be helpful to consider $\displaystyle \int_{a}^{b} \surd x dx$ for suitable a and b
Let:Originally Posted by Natasha1
$\displaystyle
S_n= 1 + \surd2 + \surd3 + 2 + \surd5 + \surd6 + ..... + \surd n
$
Then as $\displaystyle \{\sqrt{r}, r=1,2, \dots,\ n\}$ is an increasing sequence:
$\displaystyle
\int_{x=1}^{n+1} \sqrt{x-1} dx<S_n<\int_{x=1}^{n+1} \sqrt{x} dx
$
So doing the integrals gives:
$\displaystyle
\frac{2}{3}(n)^{3/2}<S_n<\frac{2}{3} ((n+1)^{3/2}-1)
$
RonL