Part 1/4:
Understanding the Ratio Test for Convergence
In mathematical analysis, the convergence of series plays a crucial role, particularly in determining the sum of infinite sequences. One powerful tool that mathematicians often use to analyze the convergence of series is the Ratio Test. In this article, we will explore a specific application of the ratio test to the series defined by the summation of ( \frac{1}{n!} ).
Exploring the Series ( \sum_{n=1}^{\infty} \frac{1}{n!} )
Consider the series given by ( a_n = \frac{1}{n!} ). To determine whether this series converges or diverges, we can apply the Ratio Test. The foundational procedure of the ratio test involves analyzing the limit of the ratio of successive terms in the series.