Question

1^p+2^p+3^p+.....+n^p

一道求极限的问题
n→无穷时,求极限(1^p+2^p+3^p+...+n^p)/n^(p+1)
(p>0)

Answer 1

lim(1^p+2^p+3^p+...+n^p)/n^(p+1)
=lim((1/n)^p+(2/n)^p+(3/n)^p+...+(n/n)^p)*1/n
=∫(0,1)x^pdx
=1/(p+1)x^(p+1)|(0,1)
=1/(p+1)