Find the value of `underset(r = 1)overset(10)sum underset(s = 1)overset(10)sum tan^(-1) ((r)/(s))`

overset(10)underset(i=3)sum5

Let n be a positive integer and (1+x)^(n)+C_(0)+C_(1)x+C_(2)x^(2)+C_(3)x^(3)+ . . .+C_(r)x^(r)+ . . .+C_(n-1)x^(n-1)+C_(n)x^(n) Where C_(r) stands for .^(n)C_(r) , then Q. The values of underset(r=0)overset(n)(sum)underset(s=0)overset(n)(sum)(C_(r)+C_(S)) is

let underset(r=0)overset(2010)(sum)a_(r)x^(r)=(1+x+x^(2)+x^(3)+x^(4)+x^(5))^(402) and underset(r=0)overset(2010)(sum)a_(r)=a, then the value of ((underset(r=0)overset(2010)(sum)r.a_(r))/(underset(r=0)overset(2010)(sum)a_(r))) is equal to____.

Let n be a positive integer and (1+x)^(n)+C_(0)+C_(1)x+C_(2)x^(2)+C_(3)x^(3)+ . . .+C_(r)x^(r)+ . . .+C_(n-1)x^(n-1)+C_(n)x^(n) Where C_(r) stands for .^(n)C_(r) , then Q. The value of underset(r=0)overset(n)(sum)underset(s=0)overset(n)(sum),C_(r),C_(S) is

Find the underset(k=1)overset(oo)sumunderset(n=1)overset(oo)sumk/(2^(n+k)) .

The value of cot (underset(n=1)overset(23)sum cot^(-1) (1 + underset(k=1)overset(n)sum 2k)) is

Find the value of underset(r = 0) overset(oo)sum tan^(-1) ((1)/(1 + r + r^(2)))

Find the value underset(n rarr oo)("lim") underset(k =2)overset(n)sum cos^(-1) ((1 + sqrt((k -1) k(k + 1) (k + 2)))/(k(k + 1)))

Let x_(1) lt x_(2) lt x_(3) lt x_(4) lt x_(5) and y_(1) lt y_(2) lt y_(3) lt y_(4) lt y_(5) are in AP such that underset(I = 1)overset(5)(sum) x_(i) = underset(I = 1)overset(5)(sum) y_(i) = 25 and underset(I =1)overset(5)(II) x_(i) = underset(I = 1)overset(5)(II) y_(i) = 0 then |y_(5) - x_(5)| A. 5//2 B. 5 C. 10 D. 15

Prove that underset(rles)(underset(r=0)overset(s)(sum)underset(s=1)overset(n)(sum))""^(n)C_(s) ""^(s)C_(r)=3^(n)-1 .