"Rules of thumb" to decide which convergence test is most appropriate Are there some general "rules of thumb" that people, who lack enough experience or intuition, could apply to decice which convergence test (such as the ratio test, the nth root test, or Raabe's test etc) is most appropriate to test the absolute convergence (or find the radius of convergence) of a series 
$$ \sum_{n=1}^{\infty} = \left|a_n\right|$$
just from looking at the definition of its elements $a_n$?
If such general rules exist, what are the underlying (deeper mathematical) motivations for them?
 A: Basically just what Michael Lugo said: do the easiest test first, then the second easiest, the third easiest and so on. I was taught [in undergraduate physics (credit to my lecturer)] to do these three tests in the order laid out:
The ratio test for convergence (d'Alembert test):
Assume $a_{n} > 0$ for all $n$ . Let $R = \lim_{n \to \infty} \frac{a_{n+1}}{a_{n}}$.
If $R < 1$, the series $\sum_{n=n_{0}}^{\infty} a_{n}$ converges.
If $R > 1$, the series $\sum_{n=n_{0}}^{\infty} a_{n}$ does not converge.
If $R = 1$, the series $\sum_{n=n_{0}}^{\infty} a_{n}$ may or may not converge.
The root test for convergence (Cauchy test):
Assume $a_{n} > 0$ for all $n$ . Let $Q = \lim_{n \to \infty}$ $(a_{n})^{1/n}$.
If $Q < 1$, the series $\sum_{n=n_{0}}^{\infty} a_{n}$ converges.
If $Q > 1$, the series $\sum_{n=n_{0}}^{\infty} a_{n}$ does not converge.
If $Q = 1$, the series $\sum_{n=n_{0}}^{\infty} a_{n}$ may or may not converge.
The integral test for convergence:
Assume that $f(x)$ is a monotonically decreasing and positive function on the interval $N \le$ x < $\infty$, and let $a_{n} = f(n)$ for integer $n$.
If $\int_{N}^{\infty} f(x) dx$ is finite, the series $\sum_{n=n_{0}}^{\infty} a_{n}$ converges.
If $\int_{N}^{\infty} f(x) dx$ is infinite, the series $\sum_{n=n_{0}}^{\infty} a_{n}$ does not converges.
Also, sometimes if you have an easy looking sum, you can tell by inspection.
Another note: always always take the limit to infinity; testing the first $10^{10^{10}}$ terms, you haven't proved that it converges.
