Final answer:
The binomial expansion of the expression (1+x)−3 is an infinite series that involves the sum of terms with progressively higher powers of x and descending powers of 1. The series is valid for |x| < 1 to ensure convergence.
Step-by-step explanation:
The question asks us to use the binomial theorem to expand the expression (1+x)−3 and to state the range of values of x for which the expansion is valid.
The binomial theorem for negative or fractional exponents can be written as:
(a + b)ⁿ = α⁰ + nαⁿ¹bⁱ + n(n-1)αⁿ²b²/2! + n(n-1)(n-2)αⁿ³b³/3! + ...
In this case, we are expanding (1+x)−3 which means 'a' is 1 and 'b' is x, and our 'n' is -3. Thus our series becomes:
- 1³+ (-3)(1)²x + (-3)(-4)(1)x²/2! + (-3)(-4)(-5)(1)x³/3! + ...
Furthermore, the series expansion is only valid for |x| < 1 to ensure convergence. Beyond that range, the series do not accurately represent the function.