Please how can i solve this one: lim(n->+00) <span title="(1/n^8)*(1^7+2^7+...+n^7)=1/8?">(1/n^8)*(1^7+2^7+...+n^7)...</span>

Question

Please how can i solve this one: lim(n->+00) (1/n^8)*(1^7+2^7+...+n^7)...

Answer 1

Notice that:

(1/n^8)∑[k = 1,n](k^7) = (1/n)∑[k = 1,n](k/n)^7

Is a Riemann sum of the function f(x) = x^7 over the interval [0,1], so:

lim (1/n^8)∑[k = 1,n](k^7) = lim (1/n)∑[k = 1,n](k/n)^7 =

= ∫[0,1](x^7)dx = 1/8