List of commonly occurring infinite sequences? I'm looking for an online list of commonly occurring convergent infinite sequence, such as

$n^a$ (converges to $0$ if $a &lt; 0$, to $1$ if $a = 0$, and to $\infty$ if $a &gt; 0$)
$\sqrt[n]{n}$ (converges to $1$)
$\sqrt[n]{n!}$ (converges to $\infty$)
$(1 + \frac{x}{n})^n$ (converges to $e^x$)

For comparison, here is a list of commonly occurring Maclaurin series, and here is a list of commonly occurring derivatives. I'm looking for a similar list, only for sequences.
$a^n$ to $0,1,\infty$ or does-not-exist is possibly the most important.
Consider $a_n=\frac{n}{\ln n\cdot \pi(n)}$. As usual $\pi(n)$ is the number of primes not exceeding $n$. It is known that $a_n\rightarrow 1$ and this fact is known as the Prime number theorem.