Question

Find a power series representation for the function.

f(x) = x6 tan−1(x3)
f(x) =
[infinity] n = 0

Answer 1

The power series representation for the function f(x) = x^6 arctan(x^3) is found by using the power series expansion of the arctangent function and then substituting x^3 into the series and multiplying by x^6.

Step-by-step explanation:

The student is asking for a power series representation for the function f(x) = x6 tan−1(x3). To find this, we can use the power series expansion of the arctangent function, which is given by:

tan−1(x) = ∑ (from n=0 to ∞) ((−1)n x2n+1) / (2n+1)

Then, substituting x3 into the series and multiplying it by x6, we can obtain the power series for the given function. Each term in the series will be of the form (−1)n x6+2(3n)+3 / (2n+1), which simplifies to (−1)n x6n+9 / (2n+1). Thus, the power series representation of the function is:

f(x) = ∑ (from n=0 to ∞) ((−1)n x6n+9 / (2n+1))