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let $ a $ real number and let $u_{n}=n^a$, let p a positive integer larger then $a$, help me to prove that the limit of $ (\Delta^p)_n= \sum_{i=0}^p (-1)^{p-i}\binom {p} {i}u_{n+i}$ is $0$ when n goes to infinity, $\Delta$ is the euler transform defined by : $ \Delta(u)_{n}= u_{n+1}-u_{n}$ and $\Delta^p=\Delta o\Delta^{p-1}$ Thank you very much

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but n^a isn't a polynomial – mathfan Jun 30 '11 at 19:35
I accidentally read it as 'a' being a positive integer, and deleted my comment after realizing my error. Sorry about that. – Jyrki Lahtonen Jun 30 '11 at 19:37

Using, for example, the fact that differences can be written down as integrals, $$ \Delta f(x)=f(x+\Delta x)-f(x)=\int_0^{\Delta x}f'(x+y_1)\, dy_1, $$ $$ \Delta^2 f(x)=f(x+2\Delta x)-2f(x+\Delta x)+f(x)=\int_0^{\Delta x}\int_0^{\Delta x}f''(x+y_1+y_2)\,dy_1dy_2, $$ etc and the mean value theorem.

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+1 Very nice :-) – Jyrki Lahtonen Jun 30 '11 at 20:45

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