The value of `Sigma_(n=1)^(3) tan^(-1)(1)/(n)` is

1.	The value of `Sigma_(n=1)^(3) tan^(-1)(1)/(n)` is
Answer» We have `underset(n=1)overset(3)Sigma tan^(-1)(1)/(n)=tan^(-1)+tan^(-1)1/2+tan^(-1)1/3` `rarr underset(n=1)overset(3)Sigma tan^(-1)(1)/(n)=(pi)/(4)O+tan^(-1)((1/2+1/3)/(1-1/2xx1/3))` `rarrunderset(n=1)overset(3)Sigma tan^(-1)(1)/(n)=(pi)/(4)+tan^(-1)1=(pi)/(4)+(pi)/(4)=(pi)/(2)`

Discussion

`tan^(-1)[(cosx)/(1+sinx)]`is equal to`pi/4-x/2,forx in (-pi/2,(3pi)/2)``pi/4-x/2,forx in (-pi/2,pi/2)``pi/4-x/2,forx in (-pi/2,pi/2)``pi/4-x/2,forx in (-(3pi)/2,pi/2)`
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