Question

A function f has derivatives of all orders for −1(a) Show that the first four nonzero terms of the Maclaurin series for f are x − x²/2 + x³/3 − x⁴/4, and write the general term of the Maclaurin series for f.

a) x - x²/2 + x³/3 - x⁴/4
b) x + x²/2 + x³/3 + x⁴/4
c) x - x²/4 + x³/9 - x⁴/16
d) x + x²/4 + x³/9 + x⁴/16

Answer 1

The Maclaurin series for the function f is x - x²/2 + x³/3 - x⁴/4, with the general term (-1)^(n+1)/n.

Step-by-step explanation:

The Maclaurin series is a special case of the Taylor series, which represents a function as an infinite sum of terms. To find the Maclaurin series for a function, we first need to find the derivatives of that function.

In this case, the function f has derivatives of all orders at x = 0. We can find the first four nonzero terms of its Maclaurin series by evaluating the function and its derivatives at x = 0.

To find the first term, we evaluate f(0) = 0. To find the second term, we evaluate f'(0) = 1. To find the third term, we evaluate f''(0) = -1/2. And to find the fourth term, we evaluate f'''(0) = 1/3.

So, the first four nonzero terms of the Maclaurin series for f are x - x²/2 + x³/3 - x⁴/4. The general term of the Maclaurin series for f can be found by using the pattern of the coefficients: (-1)^(n+1)/n, where n is the term number starting from 1.