Power Series Expansion for General Exponential Function

Theorem

Let $a \in \R_{> 0}$ be a (strictly) positive real number.

Then:

$\ds \forall x \in \R: \, $

$\ds a^x$

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac {\paren {x \ln a}^n} {n!}$

$\ds $

$=$

$\ds 1 + x \ln a + \frac {\paren {x \ln a}^2} {2!} + \frac {\paren {x \ln a}^3} {3!} + \cdots$

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 $x$.

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Series for Exponential and Logarithmic Functions: $20.16$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 22$: Taylor Series: Series for Exponential and Logarithmic Functions: $22.16.$