_ n n^4 [ 1 n^5 + 1 (n^2+1 )^ 5 2 + 1 (n^2+4 )^ 5 2 + 1 (n^2+9 )^ 5 2 + + ]=

Step-by-step solution

Since, _ n n^4 [ 1 n^5 + 1 (n^2+1 )^ 5 2 + 1 (n^2+4 )^ 5 2 + .+ 1 (n^2+n^2 )^ 5 2 ]= _ n n^4 _ r=0 ^n 1 (n^2+r^2 )^ 5 2 = _ n 1 n _ r=0 ^n n^5 (n^2+r^2 )^ 5 2 = _ n 1 n _ r=0 ^n 1 (1+ r^2 n^2 )^ 5 2 = ₀^1 1 (1+x^2 )^ 5 2 d x Let I= ₀^1 1 (1+x^2 )^ 5 2 d x Let =x ^2 d =d x I= ₀^ 4 ^2 d (1+ ^2 )^ 5 2 = ₀^ 4 1 ^3 d = ₀^ 4 ^3 d = ₀^ 4 (1- ^2 ) d Let =t d =d t I = ₀^ 1 2 (1-t^2 ) d t= (t- t^3 3 )₀^ 1 2 = ( 1 2 - 1 2 2 3 )-0 I= 6-1 6 2 = 5 6 2