$$\mathop {\lim }\limits_{n \to \infty } {1 \over {{2^n}}}\left( {{1 \over {\sqrt {1 - {1 \over {{2^n}}}} }} + {1 \over {\sqrt {1 - {2 \over {{2^n}}}} }} + {1 \over {\sqrt {1 - {3 \over {{2^n}}}} }} + \,\,...\,\, + \,\,{1 \over {\sqrt {1 - {{{2^n} - 1} \over {{2^n}}}} }}} \right)$$ is equal to

Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Then the value of the integral $$\int_{-3}^{101}\left([\sin (\pi x)]+e^{[\cos (2 \pi x)]}\right) d x$$ is equal to

For any real number $$x$$, let $$[x]$$ denote the largest integer less than equal to $$x$$. Let $$f$$ be a real valued function defined on the interval $$[-10,10]$$ by $$f(x)=\left\{\begin{array}{l}x-[x], \text { if }[x] \text { is odd } \\ 1+[x]-x, \text { if }[x] \text { is even } .\end{array}\right.$$ Then the value of $$\frac{\pi^{2}}{10} \int_{-10}^{10} f(x) \cos \pi x \,d x$$ is :

$$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{r \over {2{r^2} - 7rn + 6{n^2}}}} $$ is equal to :