Determine whether the series is divergent or convergent

Question

johnnydicamillo · Answer

$$\sum_{n =1 }^{\infty} \frac{ n }{ 1 + \sqrt{n} }$$

johnnydicamillo · Answer

actually this is a sequence of partial sums

johnnydicamillo · Answer

so I take the limit as n goes to infinity right

misty1212 · Answer

hi!!

misty1212 · Answer

eyeball it 
the degree of the numerator is larger than the degree of the denominator, so the terms don't even to go zero

johnnydicamillo · Answer

so divergent because n = infinity

johnnydicamillo · Answer

and not 0

misty1212 · Answer

$$\lim_{n\to\infty}\frac{n}{1+\sqrt{n}}=\infty$$
the terms get larger and larger
so since the terms do not go to zero, there is no way the sum converges
you are adding larger and larger numbes

johnnydicamillo · Answer

awesome thanks!

misty1212 · Answer

$$\color\magenta\heartsuit$$

misty1212 · Answer

btw it is not enough for the terms to go to zero
they have to go to zero fast enough 
but since these terms do not even go to zero, no other consideration need apply

kainui · Answer

I totally agree with @misty1212 on doing the eyeball test. Also, you can do L'Hopital's rule to evaluate the limit.