Find the first four non-zero terms of the Taylor series at x = 0 of $f ( x ) = \sin 2 x$ .
Is $f ( x ) = 2 x - \frac { 8 x ^ { 3 } } { 3 ! } + \frac { 32 x ^ { 5 } } { 5 ! } - \frac { 128 x ^ { 7 } } { 7 ! }$ the correct answer?

Question

Quizplus · Accepted Answer

The Taylor series expansion of $\sin x$ at $x = 0$ is $x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$. For $\sin 2x$, replace $x$ with $2x$ in the series: $2x - \frac{(2x)^3}{3!} + \frac{(2x)^5}{5!} - \frac{(2x)^7}{7!}$, which simplifies to $2x - \frac{8x^3}{3!} + \frac{32x^5}{5!} - \frac{128x^7}{7!}$.

Find the First Four Non-Zero Terms of the Taylor Series f(x)=sin⁡2xf ( x ) = \sin 2 xf(x)=sin2x

Find the First Four Non-Zero Terms of the Taylor Series $f ( x ) = \sin 2 x$