Solutions - 0240-01

(a)

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

$I=[-1/5,+1/5)$

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(b)

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

$I=(-\infty ,\infty )$