Sum of Series Involving Factorials

Find the sum of the series $\sum_{n=0}^{\infty} \frac{1}{n!}$.

This problem involves evaluating an infinite series whose terms are defined by a factorial in the denominator. Understanding this problem involves a deep dive into the behavior of exponential functions and how they can be linked to series, notably through the framework of the Taylor series expansion. The challenge here is to recognize the series as the Taylor series expansion for a well-known function.

The key to solving problems involving series is to recognize the patterns they form, such as power series expansions for familiar functions. In this case, we are dealing with an exponential function. One way to approach this is by remembering that the exponential function can be expressed as a power series: the sum of its derivative terms divided by factorials. Thus, this problem provides an excellent opportunity to review such expansions and their fundamental role in both pure and applied mathematics.

Furthermore, infinite series like this one frequently appear in solutions to differential equations and in calculations involving growth processes, highlighting their relevance beyond theoretical mathematics and into practical applications. This problem also serves as a gateway to understanding more complex series and their convergence properties, such as absolute and conditional convergence, which are critical in many areas of advanced mathematics.