Question

Is the series convergent or divergent? $$ \sum_{n=1}^{\infty} \frac{e^{n}}{n !} $$

Short answer

The series is convergent.

Step-by-step solution

Identify the Series Type

This series is an infinite series given by \( \sum_{n=1}^{\infty} \frac{e^{n}}{n !} \). Each term of the series can be denoted as \( a_n = \frac{e^{n}}{n !} \).

Identify the Convergence Test

Given that each term in the series contains a factorial in the denominator and an exponential in the numerator, a suitable test to apply is the Ratio Test, which is helpful for series with terms of the form \( a_n = c^n / n! \).

Apply the Ratio Test

Find the limit \( L \) of \( \left| \frac{a_{n+1}}{a_n} \right| \) as \( n \) approaches infinity. Calculate:\[ \frac{a_{n+1}}{a_n} = \frac{\frac{e^{n+1}}{(n+1)!}}{\frac{e^{n}}{n!}} = \frac{e^{n+1} \cdot n!}{e^{n} \cdot (n+1)!} = \frac{e \cdot n!}{(n+1) \cdot n!} = \frac{e}{n+1}. \]Now find the limit:\[ L = \lim_{{n \to \infty}} \frac{e}{n+1} = 0. \]

Analyze the Result of the Ratio Test

The Ratio Test states that if \( L < 1 \), then the series converges. Here, since \( L = 0 \) which is less than 1, the series converges.

Key concepts

Ratio Test

The ratio test is a powerful method used to determine the convergence or divergence of an infinite series.It is incredibly useful for series where each term is a ratio of factorial expressions and exponential functions, like in the series \( \sum_{n=1}^{\infty} \frac{e^n}{n!} \).
The test examines the limit of the ratio of consecutive terms. To apply it, you find the limit \[L = \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right|\]where \( a_n \) represents the general term of the series. If \( L < 1 \), the series converges absolutely. If \( L > 1 \), or if the limit does not exist, the series diverges. Finally, if \( L = 1 \), the test is inconclusive.
In the example given, the application of the ratio test yielded \( L = 0 \), showing the series converges, as 0 is much less than 1.

Factorial and Exponential Series

Factorial and exponential functions frequently appear in series, especially in mathematical contexts involving probabilities, combinatorics, and calculus.
Factorial of a number \( n! \) is the product of all positive integers less than or equal to \( n \). It increases very rapidly as \( n \) grows. On the other hand, exponential functions like \( e^n \) also grow rapidly but are often overshadowed by factorial growth.
In a series such as \( \sum_{n=1}^{\infty} \frac{e^n}{n!} \), the factorial in the denominator grows faster than the exponential term in the numerator as \( n \) increases. This rapid growth behavior of \( n! \) leads to each term of the series shrinking to zero, contributing significantly to the convergence of the series. Such properties are central to understanding why the ratio test works effectively on these types of series.

Analysis of Series Convergence

Analyzing the convergence of a series is a crucial part of higher mathematics, providing insights into behavior over long, infinite sums.Several tests, including the ratio test, are designed to evaluate whether an infinite series converges or diverges. Convergence means that as more terms are added, the series approaches a finite number.
For the given series \( \sum_{n=1}^{\infty} \frac{e^n}{n!} \), calculating \( L \) as 0 through the ratio test gives a straightforward proof of convergence. This conclusion is reached because the ratio of successive terms tends more toward zero than toward growing larger than one.
Understanding the dynamics of series convergence involves recognizing patterns in sequences like the factorial-exponential series and employs rigorous testing to ensure valid conclusions. This analysis is key for concepts in calculus and mathematical modeling, where convergence dictates the behavior and applicability of mathematical models.