Question

Use the preliminary test to decide whether the following series are divergent or require further testing. Careful: Do not say that a series is convergent; the preliminary test cannot decide this. $$\sum_{n=1}^{\infty} \frac{n !}{n !+1}$$

Short answer

Since \( \lim_{{n \to \infty}} \frac{n!}{n!+1} = 1 \eq 0 \), the series is divergent.

Step-by-step solution

Understand the Preliminary Test

The preliminary test states that if \( \lim_{{n \to \infty}} a_n \eq 0 \) or does not exist, the series \( \sum a_n \) is divergent. This is important to remember as it will guide the process.

Define the General Term

Identify the general term \(a_n\) of the series. In this case, \(a_n = \frac{n!}{n!+1}\).

Compute the Limit

Evaluate the limit of the general term as \(n\) approaches infinity: \lim_{{n \to \infty}} \frac{n!}{n!+1}\.

Simplify the Limit Expression

Simplify the limit expression. Notice that \( \lim_{{n \to \infty}} \frac{n!}{n!+1} = \lim_{{n \to \infty}} \frac{1}{1 + \frac{1}{n!}} = 1 \) because \( n! \to \infty \) as \( n \to \infty \).

Apply the Preliminary Test

Using the preliminary test, since \( \lim_{{n \to \infty}} \frac{n!}{n!+1} = 1 \), and not 0, it implies the series \( \sum_{{n=1}}^{{\infty}} \frac{n!}{n!+1} \) is divergent.

Key concepts

Convergence Tests

To determine whether a series converges or diverges, mathematicians use various convergence tests. One of the simplest and often the first test we use is the Preliminary Test (or Divergence Test). This test states: If the limit of the nth term of a series does not approach zero as n approaches infinity, the series diverges.

In formula terms, if \(\text{lim}_{{n \rightarrow \infty}} a_n eq 0\) or the limit does not exist, then the series \(\text{ \sum a_n}\) diverges. This insight helps us quickly identify certain divergent series without using more complex tests.

Keep in mind there are more convergence tests like the Ratio Test, Root Test, and Integral Test, which help evaluate convergence in more challenging scenarios.

Limit of a Sequence

Understanding the limit of a sequence is crucial in determining series convergence. The limit is the value that the terms of a sequence approach as the index (usually n) goes to infinity. For instance, in our exercise, we needed to find the limit of the sequence given by the general term \(\frac{n!}{n!+1}\).

To compute the limit, observe that as n grows very large, the factorial \(n!\) also grows extremely large. This makes the term \( \frac{1}{n!} \) negligible in the denominator:

\[\lim_{{n \rightarrow \infty}} \frac{n!}{n! + 1} = \lim_{{n \rightarrow \infty}} \frac{1}{1 + \frac{1}{n!}} = 1 \]

This means the limit does not approach zero, which according to the preliminary test, means the series diverges.

Factorials in Series

Factorials are a common part of many series and sequences. The factorial of a number n, denoted as \(n!\), is the product of all positive integers up to n. Factorials grow extremely fast. For example, \(5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120\).

In the provided exercise, we dealt with the general term \(\frac{n!}{n!+1}\). The rapid growth of factorials helps us simplify the limits involved.

Because \(n!\) grows so quickly as n increases, any additional constant in the denominator, like \(+1\), becomes insignificant, making the term simplify more easily when finding limits.

Understanding the properties of factorials will be very helpful in identifying how terms behave as n approaches infinity.

Evaluating Limits

Evaluating limits is a fundamental skill in calculus. It helps us understand the behavior of functions and sequences as the variable approaches a specific value, often infinity. When evaluating the limit of a fraction, as in our exercise, consider the behavior of the numerator and the denominator individually as n increases.

For \(\frac{n!}{n!+1}\), we simplify by finding the dominant terms in the numerator and the denominator. Here, as n approaches infinity, both the numerator \(n!\) and the denominator \(n! + 1\) grow, but since \(\frac{1}{n!}\) in \(n! + 1\) becomes very small, we focus on the term \( \frac{1}{1 + \frac{1}{n!}}\).

The limit evaluates to:

\[\lim_{{n \rightarrow \infty}} \frac{n!}{n! + 1} = 1 \]

Recognizing that the limit does not converge to zero means the series must diverge as per the preliminary test.