_ n [ 1^k+2^k+3^k+ +n^k n^ k+1 ]=

Step-by-step solution

We have, aligned & _ n [ 1^k+2^k+3^k+ +n^k n^ k+1 ] & = _ n 1 n [ 1^k+2^k+ . .+n^k n^k ] & = _ n 1 n [ ( 1 n )^k+ ( 2 n )^k+ ( 3 n )^k+ .+ ( n-1 n )^k+1 ] & = _ n 1 n _ r=0 ^ n-1 ( r n )^k+ _ n 1 n & .= ₀^1 x^k d x+0= [ x^ k+1 k+1 ]₀^1 ]= 1 k+1 & _ n [ 1^k+2^k+3^k+ .+n^k n^ k+1 ] aligned