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

Explanation

`S = underset(r = 1) overset(10)sum underset(s = 1) overset(10)sum tan^(-1) ((r)/(s))`
`:. S = underset(r = 1)overset(10)sum underset(s = 1)overset(10)sum tan^(-1) ((s)/(r))` (As r and s are independent)
On adding, we get
`2S = underset(r =1)overset(10)sum underset(s =1) overset(10)sum (tan^(-1) ((r)/(s)) + tan^(-1) ((s)/(r)))`
`rArr 2S = underset(r = 1)overset(10)sum underset(s =1) overset(10)sum (pi)/(2) = (pi)/(2) underset(r =1)overset(10)sum 10 = (100pi)/(2)`
`:. S = 25 pi`