Manipulated Taylor Series

Calculus Level 1

Determine the Taylor series centered about x = 0 x = 0 for the function

f ( x ) = 2 x 3 sin ⁡ 4 x 5 . f(x) = 2x^3\sin 4x^5.

∑ n = 0 ∞ ( − 1 ) n 2 4 n + 3 x 10 n + 5 ( 2 n + 1 ) ! \sum_{n = 0}^{\infty}(-1)^n \frac{2^{4n + 3} x^{10n + 5}}{(2n + 1)!} ∑ n = 0 ∞ ( − 1 ) n 4 2 n + 2 x 10 n + 5 ( 2 n + 1 ) ! \sum_{n = 0}^{\infty}(-1)^n \frac{4^{2n + 2} x^{10n + 5}}{(2n + 1)!} ∑ n = 0 ∞ ( − 1 ) n 2 4 n + 3 x 10 n + 8 ( 2 n + 1 ) ! \sum_{n = 0}^{\infty}(-1)^n \frac{2^{4n + 3} x^{10n + 8}}{(2n + 1)!} ∑ n = 0 ∞ ( − 1 ) n 4 2 n + 2 x 10 n + 8 ( 2 n + 1 ) ! \sum_{n = 0}^{\infty}(-1)^n \frac{4^{2n + 2} x^{10n + 8}}{(2n + 1)!}

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Andrew Ellinor by Andrew Ellinor , 33, USA

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1 solution

f ( x ) = 2 x 3 sin ⁡ 4 x 5 By Maclaurin series or Taylor series centered at x = 0 = 2 x 3 ∑ n = 0 ∞ ( − 1 ) n ( 4 x 5 ) 2 n + 1 ( 2 n + 1 ) ! = ∑ n = 0 ∞ ( − 1 ) n 2 ( 2 2 ) 2 n + 1 ⋅ x 3 ( x 5 ) 2 n + 1 ( 2 n + 1 ) ! = ∑ n = 0 ∞ ( − 1 ) n 2 4 n + 2 + 1 ⋅ x 10 n + 5 + 3 ( 2 n + 1 ) ! ⟹ f ( x ) = ∑ n = 0 ∞ ( − 1 ) n 2 4 n + 3 x 10 n + 8 ( 2 n + 1 ) ! \begin{aligned} f(x) & = 2x^3 \color{#3D99F6}{\sin 4x^5} & \small \color{#3D99F6}{\text{By Maclaurin series or Taylor series centered at }x=0} \\ & = 2x^3 \color{#3D99F6}{\sum_{n=0}^\infty (-1)^n \frac {(4x^5)^{2n+1}}{(2n+1)!}} \\ & = \sum_{n=0}^\infty (-1)^n \frac {2(2^2)^{2n+1}\cdot x^3(x^5)^{2n+1}}{(2n+1)!} \\ & = \sum_{n=0}^\infty (-1)^n \frac {2^{4n+2+1}\cdot x^{10n+5+3}}{(2n+1)!} \\ \implies f(x) & = \boxed{\displaystyle \sum_{n=0}^\infty (-1)^n \frac {2^{4n+3}x^{10n+8}}{(2n+1)!}} \end{aligned}

3 Helpful 0 Interesting 0 Brilliant 6 Confused

why 2^(4n+3) ?

Wolfgang Allendorf - 2 years, 4 months ago

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2 × 4 2 n + 1 = 2 × 2 4 n + 2 = 2 4 n + 3 2\times {\color{#3D99F6} 4^{2n+1}} = 2\times {\color{#3D99F6} 2^{4n+2}} = \boxed{2^{4n+3}} .

Chew-Seong Cheong - 2 years, 4 months ago

Nice question. Thanks Brilliant.org

Nandha Kumar - 1 year, 9 months ago

why x^(10n+8) and not x^(10n+4) ?

Eiko Yamada - 1 year, 3 months ago

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I have added two extra steps to explained.

Chew-Seong Cheong - 1 year, 3 months ago

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