SCalcET9 11.10.01 Find the Maclaurin series for $f(x)$ using the definition of a Maclaurin series. [Assume that $f$ has a power series expansion. Do not show that $R_n(x) \to 0$.] $f(x) = 3^x$ $f(x) = \sum_{n=0}^{\infty} (\quad)$ Find the associated radius of convergence $R$. $R = \quad$
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We need to find the derivatives of $f(x)$ evaluated at $x=0$. Step 2: Calculate the first few derivatives of $f(x) = 3^x$: $f(x) = 3^x$ $f'(x) = 3^x \ln(3)$ $f''(x) = 3^x (\ln(3))^2$ $f'''(x) = 3^x (\ln(3))^3$ In general, the $n$-th derivative is: $f^{(n)}(x) = Show more…
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Using Maclaurin's series, find the first five terms for the expansion of the function $f(x)=\mathrm{e}^{3 x}$ $$ \begin{array}{rlr} f(x) & =\mathrm{e}^{3 x} & f(0)=\mathrm{e}^{0}=1 \\ f^{\prime}(x) & =3 \mathrm{e}^{3 x} & f^{\prime}(0)=3 \mathrm{e}^{0}=3 \\ f^{\prime \prime}(x) & =9 \mathrm{e}^{3 x} & f^{\prime \prime}(0)=9 \mathrm{e}^{0}=9 \\ f^{\prime \prime \prime}(x) & =27 \mathrm{e}^{3 x} & f^{\prime \prime \prime}(0)=27 \mathrm{e}^{0}=27 \\ f^{\mathrm{iv}}(x) & =81 \mathrm{e}^{3 x} & f^{\mathrm{iv}}(0)=81 \mathrm{e}^{0}=81 \end{array} $$ Substituting the above values into Maclaurin's series of equation (5) gives: $$ \begin{aligned} &\mathrm{e}^{3 x}=1+x(3)+\frac{x^{2}}{2 !}(9)+\frac{x^{3}}{3 !}(27) \\ &+\frac{x^{4}}{4 !}(81)+\cdots \\ &\mathrm{e}^{3 x}=1+3 x+\frac{9 x^{2}}{2 !}+\frac{27 x^{3}}{3 !}+\frac{81 x^{4}}{4 !}+\cdots \end{aligned} $$ \text { i.e. } \mathrm{e}^{3 x}=1+3 x+\frac{9 x^{2}}{2}+\frac{9 x^{3}}{2}+\frac{27 x^{4}}{8}+\cdots
1. Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] f(x) = 9^x 2. Find the associated radius of convergence R. R=
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