Solutions - 0240-02

(a)

$$\begin{array}{c}\sin u=\sum _{n=0}^{\infty }{(-1)}^n\frac{u^{2n+1}}{(2n+1)!}\\ \\ \gg \gg \sin (3x^2)=\sum _{n=0}^{\infty }{(-1)}^n\frac{{(3x^2)}^{2n+1}}{(2n+1)!}\\ \\ \gg \gg \sum _{n=0}^{\infty }{(-1)}^n\frac{3^{2n+1}}{(2n+1)!}{x}^{4n+2}\end{array}$$

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(b)

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