I am just having fun with this question: Is this true that $\displaystyle \lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$?
I thought to change this to an integral, namely that $$S_n=\lim_{n \rightarrow \infty}\int_{1/n}^1\frac{x^8}{(\frac{a}{n^2}+(x+\frac{b}{n})^2)^4}dx=1$$
Some how this does not feel alright, but would be happy if you could give me a hint whether on whether this is alright. I will also be very thankful for any other hint, or other solutions. Many thanks.