$$\lim _\limits{n \rightarrow \infty} \frac{1}{n^{k+1}}[2^k+4^k+6^k+\ldots .+(2 n)^k]=$$

the expression $${{\int\limits_0^n {[x]dx} } \over {\int\limits_0^n {\{ x\} dx} }}$$, where $$[x]$$ and $$\{ x\} $$ are respectively integral and fractional part of $$x$$ and $$n \in N$$, is equal to

The value $$\int\limits_0^{1/2} {{{dx} \over {\sqrt {1 - {x^{2n}}} }}} $$ is $$(n \in N)$$

If $${I_n} = \int\limits_0^{{\pi \over 2}} {{{\cos }^n}x\cos nxdx} $$, then I$$_1$$, I$$_2$$, I$$_3$$ ... are in