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Today's calculation of Integral 164

Source: Nara Women's Universityã€€entrance exam/Science

October 21, 2006

calculusintegrationlimitcalculus computations

Problem Statement

For positive integers

n,

let

S_{n}=\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\cdots\cdots+\frac{1}{\sqrt{n}},\ T_{n}=\frac{1}{\sqrt{1+\frac{1}{2}}}+\frac{1}{\sqrt{2+\frac{1}{2}}}+\cdots\cdots+\frac{1}{\sqrt{n+\frac{1}{2}}}.

Find

\lim_{n\to\infty}\frac{T_{n}}{S_{n}}.