Which of the following is a power series representation for the function $f\left(x\right)=x^4\arctan \left(x^3\right)$ centered at $x=0$?

• A. $\sum _{n=0}^{\infty }\frac{{(-1)}^n{x}^{3n+4}}{n+1}$
• B. $\sum _{n=0}^{\infty }\frac{{(-1)}^n{x}^{6n+4}}{2n+1}$
• C. $\sum _{n=0}^{\infty }\frac{x^{4n+3}}{2n+1}$
• D. $\sum _{n=0}^{\infty }\frac{x^{n+4}}{2n+1}$

Answer: (B)

Verified step by step guidance

1. Step 1: Recall the general formula for the Taylor series expansion of arctan(x) centered at x = 0. The series is given by: arctan(x) = sum_{n=0}^{∞} (-1)^n x^(2n+1) / (2n+1).
2. Step 2: Substitute x^3 into the arctan(x) series to find the series representation for arctan(x^3). Replace x with x^3 in the formula: arctan(x^3) = sum_{n=0}^{∞} (-1)^n (x^3)^(2n+1) / (2n+1).
3. Step 3: Simplify the powers of x in the series. Since (x^3)^(2n+1) = x^(3(2n+1)), the series becomes: arctan(x^3) = sum_{n=0}^{∞} (-1)^n x^(6n+3) / (2n+1).
4. Step 4: Multiply the series representation of arctan(x^3) by x^4 to account for the given function f(x) = x^4 arctan(x^3). This results in: f(x) = x^4 * sum_{n=0}^{∞} (-1)^n x^(6n+3) / (2n+1).
5. Step 5: Combine the powers of x. Since x^4 * x^(6n+3) = x^(6n+4), the final series representation for f(x) becomes: f(x) = sum_{n=0}^{∞} (-1)^n x^(6n+4) / (2n+1).