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Today's calculation of Integral 842

Source: 2012 Yamanashi University entrance exam/Life and Environment

August 5, 2012
calculusintegrationtrigonometrylimitcalculus computations

Problem Statement

Let Sn=∫0πsin⁡nx dx (n=1, 2, , ⋯ ).S_n=\int_0^{\pi} \sin ^ n x\ dx\ (n=1,\ 2,\ ,\ \cdots). Find lim⁡n→∞nSnSn+1.\lim_{n\to\infty} nS_nS_{n+1}.
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