问题297: an=∫x^n√1+x²， 则lim n+1an+1/an=？

$$ \begin{aligned} a_n & =\int_0^{\frac{1}{n}} x^n \cdot\left(1+\frac{1}{2} x^2+o\left(x^2\right)\right) d x \\ & =\int_0^{\frac{1}{n}} x^n+\frac{1}{2} x^{n+2}+o\left(x^{n+2}\right) d x \\ & =\left.\left[\frac{1}{n+1} x^{n+1}+\frac{1}{2(n+3)} x^{n+3}+o\left(x^{n+3}\right)\right]\right|_o ^{\frac{1}{n}} \\ & =\frac{1}{(n+1) n^{n+1}}+\frac{1}{2(n+3) n^{n+3}}+o\left(\frac{1}{n^{n+3}}\right) \sim \frac{1}{(n+1) n^{n+1}} \end{aligned} $$

$$ \begin{aligned} & \lim _{n \rightarrow \infty} \frac{(n+1) a_{n+1}}{a_n} \\ = & \lim _{n \rightarrow \infty} \frac{(n+1) \cdot \frac{1}{(n+2)(n+1)^{n+2}}}{\frac{1}{(n+1) n^{n+1}}} \\ = & \lim _{n \rightarrow \infty} \frac{(n+1)}{(n+2)\left(1+\frac{1}{n}\right)^{n+1}} \\ = & \frac{1}{e} \end{aligned} $$

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