Divergent series question

1. Dec 23, 2005

eljose

let be the divergent series:

$$1^p+2^p+3^p+.....................+N^p=S(N)$$ with p>0 my

question is..how i would prove that this series S would diverge in the form:

$$S(N)=N^{p+1}/p+1$$ N--->oo

for the cases P=1,2,3,... i can use their exact sum to prove it but for the general case i can not find any prove..perhaps i should try Euler sum formula ..are the divergent series S(N) equal to the integral:

$$\int_{0}^{\infty}dxx^{p}$$ they both diverge in the same way.

Last edited: Dec 23, 2005
2. Dec 23, 2005

StatusX

Try approximating the integral:

$$\int_0^1 x^p dx$$

with strips of width 1/N.

3. Dec 28, 2005

shmoe

You still seem to be having trouble distinguishing between an equality and an asymptotic.

You are apparently trying to show that what you've called S(N) is asymptotic to $$N^{p+1}/(p+1)$$? Just compare S(N) with the integrals

$$\int_0^N x^p dx$$ and $$\int_1^{N+1} x^p dx$$.

Euler-Maclaurin summation will work as well, but is not needed for this asymptotic.