Power Series Expansion for General Exponential Function
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Theorem
Let $a \in \R_{> 0}$ be a (strictly) positive real number.
Then:
Then:
\(\ds \forall x \in \R: \, \) | \(\ds a^x\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {x \ln a}^n} {n!}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 + x \ln a + \frac {\paren {x \ln a}^2} {2!} + \frac {\paren {x \ln a}^3} {3!} + \cdots\) |
Proof
By definition of a power to a real number:
- $a^x = \map \exp {x \ln a}$
As $x \ln a$ is itself a real number, we can use Power Series Expansion for Exponential Function:
\(\ds \forall x \in \R: \, \) | \(\ds \exp x\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 + x + \frac {x^2} {2!} + \frac {x^3} {3!} + \cdots\) |
substituting $x \ln a$ for $x$.
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Series for Exponential and Logarithmic Functions: $20.16$