Limit of sums and products 

Is there a convenient way to calculate sums like these so you can evaluate the limit? It seems like in most cases you just need to know what it adds up to.

I always just remember that $$1^k + 2^k + \cdots + n^k$$ is equal to some polynomial of order $k+1$ in $n$, but to get the exact formulas, I have to look them up over and over again.
That said, the two I do remember are $$1+1+\cdots 1 = n$$ (duh) and $$1+2+\cdots + n = \frac{n(n+1)}{2}$$
