Find an expression for the general term of the Taylor series for $x \sin x=x^{2}-\frac{x^{4}}{3 !}+\frac{x^{6}}{5 !}-\frac{x^{8}}{7 !}$ .
A) $\frac{(-1)^{n-1} x^{2 n-1}}{(2 n+1) !}$
B) $(-1)^{2 n-1} \frac{x^{n+1}}{n !}$
C) $(-1)^{n} \frac{x^{2 n+2}}{(2 n+1) !}$
D) $\frac{x^{2 n}}{2 n-1}$

Question

Quizplus · Accepted Answer

The given series is a Taylor series expansion of $x \sin x$, which alternates in sign and increases the power of $x$ by 2 in each term, starting from $x^2$. The factorial in the denominator also increases by steps of 2, starting from 3!. The correct expression that matches this pattern is $(-1)^{n} \frac{x^{2 n+2}}{(2 n+1) !}$, which correctly accounts for the alternating sign, the power of $x$, and the factorial in the denominator.

Find an Expression for the General Term of the Taylor xsin⁡x=x2−x43!+x65!−x87!x \sin x=x^{2}-\frac{x^{4}}{3 !}+\frac{x^{6}}{5 !}-\frac{x^{8}}{7 !}xsinx=x2−3!x4​+5!x6​−7!x8​

Find an Expression for the General Term of the Taylor $x \sin x=x^{2}-\frac{x^{4}}{3 !}+\frac{x^{6}}{5 !}-\frac{x^{8}}{7 !}$