Convergence of the Factorial Series

Consider the series with $\frac{1}{k!}$. The question is: 'Is this a convergent series?'

When examining the convergence of a series such as the one involving the reciprocal of factorials, it's crucial to understand the behavior of factorials themselves. A factorial grows extremely rapidly, much faster than exponential functions, which means that each term in this series becomes very small very quickly as the index increases. This rapid growth in the denominator leads to a situation where the sum of the series converges. Understanding why this series converges often involves comparing it to a series you already know converges, such as the exponential series. Such comparison tests are common strategies when dealing with series. The exponential convergence hints at how the terms diminish swiftly enough for the series sum to stabilize towards a finite value. Recognizing this series' convergence involves appreciating how quickly factorial terms outpace not only linear or polynomial growth rates but also many exponential growth rates. This type of problem encourages an intuitive understanding of series behavior that can apply to other series with fast-growing terms in the denominator.