Today's calculation of Integral 562
Source: 2010 Tokyo University entrance exam/Science, Problem 2
February 25, 2010
calculusintegrationinequalitieslogarithmslimitcalculus computations
Problem Statement
(1) Show the following inequality for every natural number .
\frac {1}{2(k \plus{} 1)} < \int_0^1 \frac {1 \minus{} x}{k \plus{} x}dx < \frac {1}{2k}
(2) Show the following inequality for every natural number such that .
\frac {m \minus{} n}{2(m \plus{} 1)(n \plus{} 1)} < \log \frac {m}{n} \minus{} \sum_{k \equal{} n \plus{} 1}^{m} \frac {1}{k} < \frac {m \minus{} n}{2mn}