Chapter 8: Infinite Sequences and Series

Section 8.5: Taylor Series

Example 8.5.13

$$ Obtain the Maclaurin series for $e^{3 x^2}$ by manipulating the series for $e^x$. $$ Solution $$ Obtain the Maclaurin series for $e^{3 x^2}$  •  Write ${\ⅇ}^{3 x^2}$, being sure to use the "exponential e". •  Context Panel: Series≻Formal Power Series ${\ⅇ}^{3 x^2}$$\stackrel{\text{formal series}}{\to }$$\sum _{n\=0}^{\mathrm{\∞}}\frac{3^n{x}^{2n}}{n\!}$ Obtain the Maclauring series for $x^x$  •  Write ${\ⅇ}^x$, being sure to use the "exponential e". •  Context Panel: Series≻Formal Power Series ${\ⅇ}^x$$\stackrel{\text{formal series}}{\to }$$\sum _{n\=0}^{\mathrm{\∞}}\frac{x^n}{n\!}$ In the series for $e^x$, replace $x$ with $3 x^2$  •  Expression palette: Evaluation template •  Context Panel: Simplify≻Assuming Integer $\genfrac{}{}{0ex}{}{\frac{x^n}{n\!}}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=3 x^2}$ = $\frac{{\left(3x^2\right)}^n}{n\!}$$\stackrel{\text{assuming integer}}{\to }$$\frac{3^n{x}^{2n}}{n\!}$$$ $$

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