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Hello everybody,

I have proved that:

\(\displaystyle \displaystyle \lim_{n\to+\infty} n\int_{0}^1 x^ng(x)\mathrm{d}x=g(1)\)

with \(\displaystyle g \in\mathcal{C}^0(\left[0,1\right],\mathbb{R})\).

But I don't know how to prove this:

\(\displaystyle \displaystyle \int_0^1 x^n f(x)\mathrm{d}x=\dfrac{f(1)}{n}-\dfrac{f(1)+f'(1)}{n^2}+o_{+\infty}(\dfrac{1}{n^2})\)

with \(\displaystyle f\in\mathcal{C}^1(\left[0,1\right],\mathbb{R})\).

Thank you for your answers.

I have proved that:

\(\displaystyle \displaystyle \lim_{n\to+\infty} n\int_{0}^1 x^ng(x)\mathrm{d}x=g(1)\)

with \(\displaystyle g \in\mathcal{C}^0(\left[0,1\right],\mathbb{R})\).

But I don't know how to prove this:

\(\displaystyle \displaystyle \int_0^1 x^n f(x)\mathrm{d}x=\dfrac{f(1)}{n}-\dfrac{f(1)+f'(1)}{n^2}+o_{+\infty}(\dfrac{1}{n^2})\)

with \(\displaystyle f\in\mathcal{C}^1(\left[0,1\right],\mathbb{R})\).

Thank you for your answers.

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