Final answer:
The Maclaurin polynomial for 1/(1 + x²) is found by substituting –x² for x in the series 1 + x + x² + x³ + ..., giving the series 1 - x² + x´ - x¶ + ... up to the desired degree.
Step-by-step explanation:
The student has been tasked with finding the Maclaurin polynomial for the function 1/(1 + x²) by making a substitution in the Maclaurin series for 1/(1 - x). Recall that the Maclaurin series for 1/(1 - x) is given by:
1 + x + x² + x³ + ...
By substituting –x² for x, we get the series for 1/(1 + x²):
1 - x² + x´ - x¶ + ...
This is valid for –x² where –x² < 1, or equivalently, –x² (-1, 1). The Maclaurin polynomial refers to the finite sum approximation of this series. For example, up to the fourth degree, it would be:
1 - x² + x´