问题及解答

求极限

\[
\lim_{n\rightarrow\infty}\biggl[\frac{1}{n+1}+\frac{1}{(n^2+1)^{1/2}}+\cdots+\frac{1}{(n^n+1)^{1/n}}\biggr]
\]

[Hint] 使用夹逼准则.

注意到

\[
\frac{1}{n+1}=\frac{1}{((n+1)^i)^{1/i}} < \frac{1}{(n^i+1)^{1/i}} < \frac{1}{(n^i)^{1/i}}=\frac{1}{n},
\]

因此

\[
\frac{n}{n+1} < \sum_{i=1}^{n}\frac{1}{(n^i+1)^{1/i}} < 1,
\]

故

\[
\lim_{n\rightarrow\infty}\sum_{i=1}^{n}\frac{1}{(n^i+1)^{1/i}}=1.
\]