%5E-1-expansion.jpeg "(e^x-1)^-1 Expansion")
The Expansion of (e^x-1)^-1
The expression (e^x-1)^-1 is a significant one in mathematics, particularly in the field of calculus and exponential functions. In this article, we will explore the expansion of this expression and its applications.
Binomial Expansion
To expand (e^x-1)^-1, we can use the binomial theorem, which states that:
$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
where n is a positive integer, and $\binom{n}{k}$ is the binomial coefficient.
Applying the Binomial Theorem
Let's denote a = e^x and b = -1. Then, we can rewrite the expression as:
$(e^x-1)^{-1} = (a+b)^{-1}$
Using the binomial theorem, we can expand this expression as:
$(e^x-1)^{-1} = \sum_{k=0}^\infty \binom{-1}{k} e^{x(-1-k)} (-1)^k$
Simplifying the Expression
Simplifying the expression, we get:
$(e^x-1)^{-1} = \sum_{k=0}^\infty \frac{(-1)^k}{k!} e^{-kx}$
This is the expansion of (e^x-1)^-1 in terms of an infinite series.
Applications
This expansion has several applications in mathematics and physics, including:
1. Exponential Series
The expansion of (e^x-1)^-1 is closely related to the exponential series:
$e^x = \sum_{k=0}^\infty \frac{x^k}{k!}$
2. Calculus
This expansion is useful in calculus, particularly in the study of infinite series and integration.
3. Probability Theory
The expansion of (e^x-1)^-1 has applications in probability theory, particularly in the study of Poisson processes.
Conclusion
In this article, we have explored the expansion of (e^x-1)^-1 using the binomial theorem. This expansion has significant applications in various fields of mathematics and physics.
