Equivalence of Definitions of Real Exponential Function/Limit of Sequence implies Power Series Expansion
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Theorem
The following definition of the concept of the real exponential function:
As the Limit of a Sequence
The exponential function can be defined as the following limit of a sequence:
- $\exp x := \ds \lim_{n \mathop \to +\infty} \paren {1 + \frac x n}^n$
implies the following definition:
As a Power Series Expansion
The exponential function can be defined as a power series:
- $\exp x := \ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$
Proof
Let $\exp x$ be the real function defined as the limit of the sequence:
- $\exp x := \ds \lim_{n \mathop \to \infty} \paren {1 + \frac x n}^n$
From the General Binomial Theorem:
\(\ds \paren {1 + \frac x n}^n\) | \(=\) | \(\ds 1 + x + \frac {n \paren {n - 1} x^2} {2! \ n^2} + \frac {n \paren {n - 1} \paren {n - 2} x^3} {3! \ n^3} + \cdots\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^0} {0!} + \frac {x^1} {1!} + \paren {\frac {n - 1} n} \frac {x^2} {2!} + \paren {\frac {\paren {n - 1} \paren {n - 2} } {n^2} } \frac {x^3} {3!} + \cdots\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 0\) | \(=\) | \(\ds \paren {1 + \frac x n}^n - \paren {\frac {x^0} {0!} + \frac {x^1} {1!} + \paren {\frac {n - 1} n} \frac {x^2} {2!} + \paren {\frac {\paren {n - 1} \paren {n - 2} } {n^2} } \frac {x^3} {3!} + \cdots}\) |
From Power over Factorial, this converges to:
- $\exp x - \paren {\dfrac {x^0} {0!} + \dfrac {x^1} {1!} + \dfrac {x^2} {2!} + \dfrac {x^3} {3!} + \cdots} = 0$
as $n \to +\infty$.
\(\ds \leadsto \ \ \) | \(\ds \exp x\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}\) |
$\blacksquare$