MathDB

Express the following the limit values in terms of a definite integral and find them. (1)

I=\lim_{n\to\infty}\frac{1}{n}\ln \left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)\cdots\cdots \left(1+\frac{n}{n}\right).

(2)

J=\lim_{n\to\infty}\frac{1}{n^{2}}(\sqrt{n^{2}-1}+\sqrt{n^{2}-2^{2}}+\cdots\cdots+\sqrt{n^{2}-n^{2}}).

(3)

K=\lim_{n\to\infty}\frac{1}{n^{3}}(\sqrt{n^{2}+1}+2\sqrt{n^{2}+2^{2}}+\cdots+n\sqrt{n^{2}+n^{2}}).

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