Chapter 8: Infinite Sequences and Series

Section 8.5: Taylor Series

Example 8.5.15

$$ Obtain the Maclaurin series for $x\/\left(1-x^2\right)$ from an appropriate geometric series. $$ Solution $$ From the formula for the sum of a geometric series with $r\=x$, obtain $\frac{1}{1-x}\=\sum _{n\=0}^{\infty }{x}^n$. In the right-hand side, replace $x$ with $x^2$, and multiply the resulting series termwise by $x$. The result is then $\frac{x}{1-x^2}\=\sum _{n\=0}^{\infty }{x}^{2 n\+1}$. A Maple corroboration is obtained as follows. $$ $\mathrm{convert}\left(x\/\left(1-x^2\right)\,\mathrm{FormalPowerSeries}\right)$ $\sum _{k\=0}^{\mathrm{\∞}}{x}^{2k\+1}$ $$

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