- #1
diredragon
- 323
- 15
Homework Statement
Find the limit of the following sequence:
##L_2 = \lim_{n \rightarrow +\infty} \frac {\sum_{k=0}^n (2k - 1)^p}{n^{p+1}}##
Homework Equations
3. The Attempt at a Solution [/B]
Seeing that ##\lim_{n \rightarrow +\infty} n^{p+1} = + \infty ## i can apply the Stolz theorem. (Is something more necessary alongside it being divergent?).
The upper sum when used in Stolz theorem yields just ##(2n+1)^{p}## because the rest of the sum is subtracted in form ##(n+1) - (n)##. I get
##\lim_{n \rightarrow +\infty} \frac{(2n+1)^p}{(n+1)^{p+1} - n^{p+1}}##
##\lim_{n \rightarrow +\infty} \frac{(2n+1)^p}{(n+1)^p + (n+1)^{p-1}n + (n+1)^{p-2}n^2 + ... + n^p}##
And I am stuck on how to further simplify this. Could get a hint from someone that knows?