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.

Knowing that the sum of the $n$ first $k^{th}$ powers is asymptotic to $\dfrac{n^{k+1}}{k+1}$, you derive that the sum of the powers of the even numbers from $2$ to $2n$ is $\sim2^k\dfrac{n^{k+1}}{k+1}=\dfrac12\dfrac{(2n)^{k+1}}{k+1}$ and that of the odd numbers from $1$ to $2n-1$ is the difference, $\sim\dfrac{(2n)^{k+1}}{k+1}-\dfrac12\dfrac{(2n)^{k+1}}{k+1}=\dfrac12\dfrac{(2n)^{k+1}}{k+1}$.
This gives the limits $1$ and $\dfrac34$.
