Find the Taylor series centered at $x=0$ (i.e.the Maclaurin series)for $\sin \left(x^{2}\right)$ .
A) $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{4 i+3}}{(4 i+1) !}$
B) $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{4 i+2}}{(2 i+1) !}$
C) $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{4 i}}{(2 i) !}$
D) $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{4 i+1}}{(4 i) !}$

Question

Quizplus · Accepted Answer

The Maclaurin series for $\sin(x)$ is $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{2i+1}}{(2i+1)!}$. Replacing $x$ with $x^2$ gives $\sum_{i=0}^{\infty} \frac{(-1)^{i} (x^2)^{2i+1}}{(2i+1)!} = \sum_{i=0}^{\infty} \frac{(-1)^{i} x^{4i+2}}{(2i+1)!}$.

Find the Taylor Series Centered At x=0x=0x=0 (IEThe Maclaurin Series)for sin⁡(x2)\sin \left(x^{2}\right)sin(x2)

Find the Taylor Series Centered At $x=0$ (IEThe Maclaurin Series)for $\sin \left(x^{2}\right)$