题目
7.(单选题,4.0分)-|||-.lim _(narrow infty )(dfrac (1)({n)^2+sqrt (1)}+dfrac (2)({n)^2+2sqrt (2)}+... +dfrac (n)({n)^2+nsqrt (n)}) =()()-|||-A ∞
题目解答
答案
:
$\lim _{n\rightarrow \infty }\sum\limits _{k=1}^{n}\dfrac {k}{n^{2}+k\sqrt {k}}=\lim _{n\rightarrow \infty }\sum\limits _{k=1}^{n}\dfrac {k}{n^{2}}+\lim _{n\rightarrow \infty }\sum\limits _{k=1}^{n}\dfrac {k}{n^{2}}\cdot \dfrac {1}{\sqrt {k}}=\int _{0}^{1}x\d x+\dfrac {2}{n}\int _{0}^{1}\dfrac {1}{\sqrt {x}}\d x=\dfrac {4}{3}$
$\dfrac {4}{3}$
$\lim _{n\rightarrow \infty }\sum\limits _{k=1}^{n}\dfrac {k}{n^{2}+k\sqrt {k}}=\lim _{n\rightarrow \infty }\sum\limits _{k=1}^{n}\dfrac {k}{n^{2}}+\lim _{n\rightarrow \infty }\sum\limits _{k=1}^{n}\dfrac {k}{n^{2}}\cdot \dfrac {1}{\sqrt {k}}=\int _{0}^{1}x\d x+\dfrac {2}{n}\int _{0}^{1}\dfrac {1}{\sqrt {x}}\d x=\dfrac {4}{3}$
$\dfrac {4}{3}$