Solutions - 0240-05

(a)

$$\begin{array}{c}\sum _{n=0}^{\infty }{(-1)}^n\frac{{\pi }^{2n+1}}{4^{2n+1}(2n+1)!}\gg \gg \sum _{n=0}^{\infty }{(-1)}^n\frac{{(\pi /4)}^{2n+1}}{(2n+1)!}\\ \\ \gg \gg \sin x{|}_{x=\pi /4}\gg \gg \frac{\sqrt{2}}{2}\end{array}$$

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(b)

$$\begin{array}{c}\sum _{n=0}^{\infty }\frac{2^{2n}}{n!}\gg \gg \sum _{n=0}^{\infty }\frac{{(4)}^n}{n!}\\ \\ \gg \gg e^x{|}_{x=4}\gg \gg e^4\end{array}$$

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(c)

$$\begin{array}{c}\sum _{n=0}^{\infty }{(-1)}^n\frac{{\pi }^{2n+2}}{3^{2n+1}(2n)!}\gg \gg \frac{{\pi }^2}{3}\sum _{n=0}^{\infty }{(-1)}^n\frac{{(\pi /3)}^{2n}}{(2n)!}\\ \\ \gg \gg \frac{{\pi }^2}{3}\cos x{|}_{x=\pi /3}\gg \gg \frac{{\pi }^2}{6}\end{array}$$