Let for each integer .

MATHEMATICS · Multi-part · AP · Calculus BC · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Let

S_n=\displaystyle\int_{0}^{\pi/2}\sin^{n}x\,dx

for each integer

n\ge 0

. (1) Derive a formula that expresses

S_{n}

in terms of

S_{n-2}

for all integers

n\ge2

. Clearly state any algebraic factors that appear in the relationship. (2) Using your result from part (a), compute an exact value for

S_{4}

. Show all steps in your reasoning. (3) Prove that

\displaystyle\frac{S_{n}}{S_{n-2}}=\frac{n-1}{n}

for every integer

n\ge2

. (4) Evaluate

\displaystyle\lim_{n\to\infty}\sqrt[n]{S_{n}}

. Justify your conclusion rigorously.