Question

Determine if the series converge or diverge.
\[\Huge\sum_{n=1}^{\infty} \dfrac{1}{2+\sqrt{n}}\]

Answer 1

I have a feel that I should use direct comparison, but I don't know how to...

Answer 2

Well, to use direct comparison, you need a suspicion... what do you suspect of this series?

Answer 3

I suspect it's divergent ^.^

Answer 4

you can ignore 2 and i feel it would be converge

Answer 5

if u imagine it as a continues function and finding the integral , you will find out tat it's infinity

Answer 6

My favourite divergent series (and probably most anybody's) is
\[\huge \sum_{n=1}^{\infty}\frac1n\]

Answer 7

So i should use integral test? @Roya

Answer 8

so it seems that it's diverge

Answer 9

No :/
It's easy to see (maybe using the Limit Comparison Test) that
\[\huge \sum_{n=1}^{\infty}\frac1{2+n}\]

Answer 10

i guess so, @PeterPan is right too

Answer 11

I tired this, for first four term, \(\dfrac{1}{n} \ge\dfrac{1}{2+\sqrt{n}}\) Should I worry about that?

Answer 12

Don't use 1/n
use its variant that I posted, 1/(2+n)
It's easier to see it that way :)

Answer 13

So, clearly
\[\huge \frac{1}{2+n}\le\frac{1}{2+\sqrt n} \ \ \ \forall n \in \mathbb{N}\]

Answer 14

Ok thank!

Answer 15

^.^