%5E-2-expansion.jpeg "(1-d)^-2 Expansion")
Understanding the Expansion of (1-d)^-2
The expression (1-d)^-2 is a common mathematical construct that often arises in fields like physics, engineering, and statistics. Expanding this expression can be achieved through a few different methods, each with its own advantages and complexities.
Using the Binomial Theorem
One approach is to utilize the binomial theorem, a powerful tool for expanding expressions of the form (x+y)^n. The binomial theorem states:
(x+y)^n = ∑_(k=0)^n (n choose k) * x^(n-k) * y^k
Where (n choose k) represents the binomial coefficient, calculated as n!/(k!(n-k)!).
To apply this to (1-d)^-2, we need to consider a few points:
- Negative exponent: The exponent -2 indicates that we are dealing with the reciprocal of (1-d)^2.
- Infinite series: The binomial theorem, when applied to negative exponents, generates an infinite series.
Therefore, we rewrite (1-d)^-2 as 1/(1-d)^2 and apply the binomial theorem to (1-d)^2:
(1-d)^2 = ∑_(k=0)^2 (2 choose k) * 1^(2-k) * (-d)^k = 1 - 2d + d^2
Taking the reciprocal and expanding the infinite series:
(1-d)^-2 = 1/(1 - 2d + d^2) = 1 + 2d + 3d^2 + 4d^3 + ...
This expansion demonstrates that the terms in the series have increasing powers of d with coefficients that are consecutive natural numbers.
Using the Geometric Series Formula
Another approach is to use the geometric series formula:
(1-x)^-1 = 1 + x + x^2 + x^3 + ... for |x| < 1
We can rewrite (1-d)^-2 as:
(1-d)^-2 = [(1-d)^-1]^2
Applying the geometric series formula to (1-d)^-1:
(1-d)^-1 = 1 + d + d^2 + d^3 + ...
Squaring this series:
(1-d)^-2 = (1 + d + d^2 + d^3 + ...)^2 = 1 + 2d + 3d^2 + 4d^3 + ...
This method provides a more direct approach to the expansion, avoiding the need to explicitly deal with negative exponents and infinite series.
Convergence and Validity
Both methods lead to the same expansion:
(1-d)^-2 = 1 + 2d + 3d^2 + 4d^3 + ...
It's important to note that this expansion is valid only for |d| < 1. For values of d outside this range, the series diverges and does not provide a meaningful representation of (1-d)^-2.
Applications
The expansion of (1-d)^-2 finds applications in various domains:
- Physics: In quantum mechanics, it appears in calculations involving perturbation theory.
- Statistics: It arises in the context of probability distributions and moment generating functions.
- Finance: It plays a role in discounted cash flow analysis.
Understanding the expansion of (1-d)^-2 provides a valuable tool for tackling various mathematical problems across diverse fields.