Question

Consider the function

1/(1−x⁴)

Write a partial sum for the power series which represents this function consisting of the first 5 nonzero terms. For example, if the series weren=0 to [infinity] 3nx2n , you would write1+3x2+32x4+33x6+34x8. Also indicate the radius of convergence.
Partial Sum:
Radius of Convergence:

Answer 1

The partial sum of the power series for 1/(1-x⁴) consisting of the first five non-zero terms is 1 + x⁴ + x⁸ + x¹⁲ + x¹⁶, and the radius of convergence for this series is 1.

Step-by-step explanation:

To find the partial sum for the power series representing the function 1/(1-x⁴), you can use the geometric series expansion. This function can be considered as a geometric series with a first term of 1 and a common ratio of x⁴. For the first five non-zero terms, the geometric series sum can be written as:

1 + x⁴ + x⁸ + x¹⁲ + x¹⁶

The radius of convergence for this power series can be found by using the ratio test. Since the series has a common ratio of x⁴, the magnitude of this ratio must be less than 1 for the series to converge:

|x⁴| < 1

This inequality is true for all x such that |x| < 1. Therefore, the radius of convergence is 1.