If $tan^{-1}\frac{n}{\pi} >\frac{\pi}{4}, $ n ∈ N, then the minimum value of n is 

4

We have

$tan^{-1} \frac{n}{\pi} >\frac{\pi}{4}$

$ ⇒ tan^{-1}\frac{n}{\pi} > tan^{-1} 1 $         $[∵\frac{\pi}{4}= tan^{-1} 1]$

$ ⇒ tan\left(tan^{-1}\frac{n}{\pi}\right) > tan \left(tan^{-1} 1\right)$     [∵ tan θ is an increasing function]

$ ⇒ \frac{n}{\pi}> 1$           $[∵ tan(tan^{-1}x)=x]$

$ ⇒n > \pi ≅ 3.14 $

$ ⇒n = 4, 5, 6 ........$

Hence, the number value of n is 4.