Question

Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{n^{n}}{n !} $$

Short answer

The series \( \sum_{n=1}^{\infty} \frac{n^{n}}{n !} \) does not give a decisive result by the Ratio Test. Further analysis is needed to assess the convergence or divergence.

Step-by-step solution

Write down the nth term and (n+1)th term of the series

The nth term of the series is \( \frac{n^{n}}{n !} \). To find the (n+1)th term, replace n with n+1, which results in \( \frac{(n+1)^{(n+1)}}{(n+1) !} \)

Compute the ratio of (n+1)th term to nth term

The ratio is then \( \frac{\frac{(n+1)^{(n+1)}}{(n+1) !}}{\frac{n^{n}}{n !}} \). This simplifies to \( \frac{(n+1)^n*(n+1)}{n^n*n} \). Factorials cancel and you can simplify the result to \( \frac{(n+1)}{n} = 1 + \frac{1}{n} \)

Apply the limit

The Ratio Test requires finding the limit as n approaches infinity, which is \( \lim_{n\to\infty} 1 + \frac{1}{n} \)

Interpret the result based on the Ratio Test

The limit is 1 which does not give us any concrete information by the Ratio Test. For the Ratio Test, if the limit is less than 1, the series is convergent; if it's greater than 1, the series is divergent. A limit of 1 is inconclusive.

Key concepts

Series Convergence

Understanding series convergence is fundamental for anyone delving into calculus or any field that involves infinite sums. A series is essentially a sum of the terms of a sequence. The concept of convergence pertains to whether the sum of these infinitely many terms approaches a finite value as more and more terms are added to the sum.

There are various tests to determine if a series converges or diverges, among which the Ratio Test is a popular method. It's particularly useful when the series involves factorials or exponential functions. The test evaluates the limit of the absolute value of the ratio of consecutive terms. If this limit is strictly less than one, the series converges absolutely. If the limit is greater than one or positive infinity, the series diverges. However, if the limit is exactly one, the test is inconclusive, which means that additional methods must be considered to determine the convergence or divergence of the series.

Factorial

The factorial, denoted by an exclamation point (!), is a function that multiplies a series of descending natural numbers. For instance, the factorial of 5, written as 5!, is calculated as
\(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

Factorials grow at an extremely rapid rate, which impacts their behavior in series and their contribution to convergence. Factorials are commonly found in the study of sequences and series, combinatorics, and other areas of advanced mathematics. When factorial terms are present in series, it's important to carefully analyze their growth compared to other parts of the terms, as they can greatly influence the series' convergence behavior. In the case of \(\frac{n^n}{n!}\), the factorial denominators grow so quickly that they often outpace the growth of numerators, leading to convergent behavior, although each series needs to be evaluated on a case-by-case basis.

Limits in Calculus

Limits are a cornerstone of calculus and are used to define continuity, derivatives, and integrals. A limit captures the behavior of a function as the input approaches a certain value. When discussing series and the Ratio Test, we look for the limit of the ratio of successive terms as n, the term number, approaches infinity.

In calculating limits, it's important to recognize forms that are indeterminate, such as \(\frac{0}{0}\) and \(\frac{\text{\textinfinity}}{\text{\textinfinity}}\), as they require further algebraic manipulation or the application of L'Hôpital's rule. However, when evaluating a series with the Ratio Test as in our example, we want to determine the behavior of the series as the term number becomes very large. If a limit leads to a constant value not equal to zero or infinity, it provides insight into the ratio between consecutive terms. Nonetheless, a limit resulting in 1 is tricky because it can be associated with both convergent and divergent series, thereby necessitating additional analysis.