Today's calculation of Integral 296
Source: 2008 Ritsumeikan University entrance exam
February 10, 2008
calculusintegrationtrigonometrycalculus computations
Problem Statement
Let a_n\equal{}\int_0^{\frac{\pi}{2} } (1\minus{}\sin t)^n\sin 2t\ dt.
(1) Find \sum_{n\equal{}1}^{\infty} a_n.
(2) Find \sum_{n\equal{}1}^{\infty} \frac{a_n}{n}.
(3) Find \sum_{n\equal{}1}^{\infty} (n\plus{}1)(a_n\minus{}a_{n\plus{}1}).