$\underset{x→∞}{\lim}\frac{2^K+4^K+6^K+....+(2n)^K}{n^{K+1}},K≠-1$ is equal to

$2^K$ $\frac{2^K}{K+1}$ $\frac{1}{K+1}$ none of these

$\frac{2^K}{K+1}$

Given sum = $\underset{x→∞}{\lim}2^K.\frac{1^K+2^K+....+n^K}{n.n^K}=2^K\underset{x→∞}{\lim}\sum\frac{1}{n}.\left(\frac{r}{n}\right)^K=2^K\int\limits_0^1x^K\,dx$ $=2^K.\left|\frac{x^{K+1}}{K+1}\right|_0^1=\frac{2^K}{K+1}$