Let $k,n \in\mathbb N$ and let $f(x)=a_{0}x^{kn}+a_{1}x^{kn-1}+\cdots+a_{kn}$. Find the limit $$\lim_{x\to+\infty}\sum_{j=0}^k (-1)^j \binom{k}{j} \sqrt[n] {f(x+k-j)}$$
I think this uses Taylor's therom?But I can't use it.
After I post I must go bed, because now is very late. Thank you.
By the way:
I think my problem is hard as this problem
If $g$ is sufficiently smooth (say, a converging power series) and $$F(x)=\sum_{i=0}^k{k\choose i}(-1)^i g(x+k-i)$$ then $$ F(x) = g^{(k)}(x+\theta) $$ with $0<\theta<k$. Therefore, we need to evaluate the $k$th derivative of $\sqrt[n]{f(x)}$. But as $f(x)=a_0x^{nk}(1+O(\frac1x))$, we have $\sqrt[n]{f(x)}=\sqrt[n]{a_0} x^k(1+O(\frac1x))$ and the $k$th derivative is $k!\sqrt[n]{a_0}(1+o(\frac1x))$ and the limit is $k!\sqrt[n]{a_0}$.
