问题及解答

设 $f\in C^2[0,1]$, 满足 $f(1)=0$, $f'(1)=a$. 证明

\[
\lim_{n\rightarrow\infty}n^2\int_0^1 x^n f(x)\mathrm{d}x=-a.
\]

\[
\begin{split}
\int_0^1 x^n f(x)dx&=\frac{1}{n+1}\int_0^1 f(x)dx^{n+1}\\
&=\frac{1}{n+1}\biggl[x^{n+1}f(x)\biggr|_{0}^{1}-\int_0^1 x^{n+1}df(x)\biggr]\\
&=-\frac{1}{n+1}\int_0^1 x^{n+1}f'(x)dx\\
&=-\frac{1}{(n+1)(n+2)}\int_0^1 f'(x)dx^{n+2}\\
&=-\frac{1}{(n+1)(n+2)}\biggl[f'(x)x^{n+2}\biggr|_{0}^{1}-\int_0^1 x^{n+2}df'(x)\biggr]\\
&=-\frac{1}{(n+1)(n+2)}\biggl[a-\int_0^1 x^{n+2}df'(x)\biggr]\\
\end{split}
\]

因此

\[
\lim_{n\rightarrow\infty}n^2\int_0^1 x^n f(x)dx=-a+\lim_{n\rightarrow\infty}\int_0^1 x^{n+2}f''(x)dx.
\]

注意到 $f\in C^2[0,1]$, 所以 存在 $M>0$, 使得 $|f''(x)|\leqslant M$. 而

\[
\lim_{n\rightarrow\infty}\int_0^1 x^{n+2}dx=0,
\]

因此

\[
\lim_{n\rightarrow\infty}\int_0^1 x^{n+2}f''(x)dx=0.
\]

从而

\[
\lim_{n\rightarrow\infty}n^2\int_0^1 x^n f(x)dx=-a.
\]