How can the infinite series for e^x be used to calculate an approximation?

The exponential function $e^x$ can be approximated using its Taylor series (specifically a Maclaurin series), which represents the function as an infinite series of powers of $x$:1

$$e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\dots$$

To calculate an approximation, you sum a finite number of these terms. Because this series converges for all real values of $x$, adding more terms generally increases the accuracy of the approximation.21

The process works as follows:
* Partial Sums: You calculate the partial sum by stopping after a specific number of terms.34
* Convergence: As you include more terms, the partial sum approaches the true value of $e^x$.34
* Efficiency: While the series converges for any $x$, the speed of convergence can vary. For values of $x$ far from zero, it may require calculating many terms to achieve a desired level of precision. In such cases, mathematicians may use alternative approaches, such as centering the series around a value $c$ near $x$ to improve the rate of convergence.2