Exercise 8.3.3

Question

\begin{enumerate}[label=(\alph*)] 
\item Using the simple identity $sin^n(x) = \sin^{n-1}(x) \sin(x)$ and the previous exercise, derive the recurrence relation
$$b_n = \frac{n-1}{n}b_{n-2}$$
for all $n \geq 2$.
\item Use this relation to generate the first three even terms and the first three odd terms of the sequence $(b_n)$.
\item Write a general expression for $b_{2n}$ and $b_{2n+1}$.
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item Apply integration-by-parts with $h = \sin^{n-1}(x)$ and $k = -\cos(x)$:
$$ \begin{aligned}
    \int^\frac{\pi}{2}_0 \sin^{n-1}(x) \sin(x) &= \left(\sin^{n-1}\frac{\pi}{2}\right)\left(-\cos \frac{\pi}{2}\right) - \left(\sin^{n-1} 0\right)\left(\cos 0\right) \
        &+ \int^\frac{\pi}{2}_0 (n-1)\sin^{n-2}(x)\cos^2(x) \
    b_n &=(n-1)\int^\frac{\pi}{2}_0 \sin^{n-2}(x) \left(1 - \sin^2(x)\right) = (n-1) b_{n-2} - (n-1)b_n \
    n b_n &= (n-1) b_{n-2} \
    b_n &= \frac{n-1}{n}b_{n-2}
\end{aligned}$$
\item Evens: $b_2 = \frac{1}{4} \pi$, $b_4 = \frac{3}{16}\pi$, $b_6 = \frac{5}{32} \pi$. Odds: $b_1 = 1$, $b_3 = \frac{2}{3}$, $b_5 = \frac{8}{15}$
\item For $n \geq 1$,
$$b_{2n} = \frac{\pi}{2}\prod^n_{i=1} \frac{2i - 1}{2i} \text{ and } b_{2n+1} = \prod^n_{i=1} \frac{2i}{2i + 1}$$
 \end{enumerate}

Exercise 8.3.3

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Exercise 8.3.3

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