Solutions - 0240-06

(a)

$$\begin{array}{c}\sum _{n=0}^{\infty }{(-1)}^n\frac{5x^{4n+2}}{(2n+1)!}\gg \gg 5\sum _{n=0}^{\infty }{(-1)}^n\frac{{(x^2)}^{2n+1}}{(2n+1)!}\\ \\ \gg \gg 5\sin (x^2)\end{array}$$

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(b)

$$\begin{array}{c}\sum _{n=0}^{\infty }\frac{{(-5x)}^{n+1}}{n+1}\gg \gg -\sum _{n=0}^{\infty }-\frac{{(-5x)}^{n+1}}{n+1}\\ \\ \gg \gg -\ln (1-(-5x))\gg \gg -\ln (1+5x)\end{array}$$

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(c)

$$\sum _{n=0}^{\infty }\frac{{(-5)}^n}{n!}\gg \gg e^x{|}_{x=-5}\gg \gg e^{-5}$$

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(d)

$$\begin{array}{c}\sum _{n=0}^{\infty }\frac{{(-1)}^n{\pi }^{2n}}{9^n(2n)!}\gg \gg \sum _{n=0}^{\infty }{(-1)}^n\frac{{(\pi /3)}^{2n}}{(2n)!}\\ \\ \gg \gg \cos x{|}_{x=\pi /3}\gg \gg \cos \frac{\pi }{3}\gg \gg \frac{1}{2}\end{array}$$