Does the series converge or diverge?
$$\sum_{1}^{infinity}\frac{\sqrt{n}}{n+1}$$

Question

jamesj · Answer

Notice that
$$ \frac{\sqrt{n}}{n+1} > \frac{1}{n+1} $$

Now does the infinite sum of the terms on the right converge?

kirbykirby · Answer

Oh ok the right one diverges (similar to the harmonic one)..  so the left one diverges by comparison test right?

jamesj · Answer

yes

kirbykirby · Answer

ok thanks :)!

anonymous · Answer

jamesj  is exactly right, but with a little practice you do this with your eyeballs.  degree of numerator is 1/2 and denominator is 1
if the degree of the  denominator does not exceed the degree of the numerator by more than one it diverges.   i will be willing to bet that is what jamesj thought (correct me if i am wrong) before he wrote the answer

kirbykirby · Answer

our prof won't allow us to "eyeball" stuff.. we have to show exactly each step -_-

jamesj · Answer

sat73 is right; I know that heuristic and used it here first.  It's a convenient rule to have.

Knowing now what the answer 'should be', I decided what was the best way to prove it. The method I suggested and you're now using is my estimation definitely the most elementary.