To find the Maclaurin series for the function f(x) = (1 + x)-4, we can utilize the binomial series expansion. The binomial series allows us to express functions in the form of (1 + x)k, where k can be any real number.
The general form of the binomial expansion is:
(1 + x)k = ∑n=0∞ C(k, n) xn, where C(k, n) = k(k-1)(k-2)…(k-n+1)/n! is the binomial coefficient.
For our specific case, we have:
k = -4
f(x) = (1 + x)-4Now, we want to expand this series using the binomial series formula:
f(x) = ∑n=0∞ C(-4, n) xnThe binomial coefficient can be calculated as:
C(-4, n) = (-4)(-5)(-6)...(-4 - n + 1)/n! = (-1)n * 4 * (4 + n - 1) * (4 + n - 2) * ... * 3 / n!We can simplify this to:
C(-4, n) = (-1)n * (4 + n – 1)! / ((n)!(3)!) = (-1)n * (4 + n – 1)(4 + n – 2)(4 + n – 3) / n!
For the first few terms (i.e. for n=0, n=1, and n=2), we get:
- For n=0: C(-4, 0) = 1
- For n=1: C(-4, 1) = -4
- For n=2: C(-4, 2) = 6
- For n=3: C(-4, 3) = -4
Thus, we can write the first few terms of the series:
f(x) ≈ 1 - 4x + 6x2 - 4x3 + ...Collecting these together, the Maclaurin series for f(x) = (1 + x)-4 is:
f(x) = ∑n=0∞ C(-4, n) xn = 1 - 4x + 6x2 - 4x3 + ...This series converges for |x| < 1.
In summary, the binomial series can provide a powerful method to derive the Maclaurin series for functions of the form (1 + x)k. By substituting the value of k and calculating the relevant coefficients, we can obtain a series that approximates the function near the point x = 0.