How to know which test to chose when proving the convergence of a serie? I need to prove the convergence of this serie : 
$$\sum\limits_{n=1}^\infty\frac{1}{\sqrt{n^2+n}}$$
I tought the easiest would be to use the ratio test, however I can't figure out how to solve. Which let me believe that I've chosen the wrong test.
I'm currently studying series and I keep facing the same issue :
How to know which test I should chose when proving the convergence of a serie ? Maybe there is a pattern to apply, or it simply comes with practice ?
 A: HINT: It behaves as
$$
\sum\limits_{n=1}^\infty\frac{1}{n}
$$
(ratio test), hence...
And in general there is no rule, how to verify the convergence of a series. Practice, practice, practice...
A: Since, $\sqrt{n^2}=n$, it makes sense to use the limit comparison test with $1/n$.
$$\lim_{n\to\infty}{1/\sqrt{n^2+n}\over1/n}=\lim_{n\to\infty}{n\over\sqrt{n^2+n}}=\lim_{n\to\infty}{1\over\sqrt{1+1/n}}=1.$$
Now, since $\sum_{n=1}^\infty\frac1n$ diverges, the given series diverges.
A: There is pattern and also practice which will make you decide which test to apply .
To know about which test to apply to which problems see video given in link below .They are by Adrisn Banner of Princeton Uiniversity  .Will certainly be helpful to you
http://www.youtube.com/watch?v=w6Yo1hYf9eM
A: All you need to know is the general behaviour of some important functions:
$$x!,e^x,x^n,\ln x$$
Generally the function which grows faster than the others is $x!$ then is $e^x$ , $x^n$ and last  is the logarithm $\ln x$ which grows too slow.
The rule is the following:
If you add,subtract,multiply or divide any combination of those two functions the faster will prevail.
For example if you want to test the convergence of $\sum \frac{1}{x^2 +\ln x}$ you know that the series behaves actually like $\sum \frac{1}{x^2}$ (which converges).
The series $\sum \frac{1}{x+e^x}$ behaves like $\sum \frac{1}{e^x}$.
But if you want to prove that it converges you have to use the ratio test.
Some hint to show you how to have an idea for the behaviour of the series. 
