Question

Use the Direct Comparison Test to determine the convergence of the given series; state what series is used for comparison. $$\sum_{n=1}^{\infty} \frac{1}{n !+n}$$

Short answer

The series converges by the Direct Comparison Test, using \( \sum \frac{1}{n!} \) for comparison.

Step-by-step solution

Identify a known convergent series

Consider a series that is known to converge and can help us with the comparison. A suitable series is \( \sum_{n=1}^{\infty} \frac{1}{n!} \). This series converges because \( n! \) grows very quickly as \( n \) increases, making \( \frac{1}{n!} \) very small.

Establish a comparison

Notice that for every term in our original series: \( n! + n > n! \). Therefore, we have: \( \frac{1}{n! + n} < \frac{1}{n!} \). This setup allows us to use the Direct Comparison Test.

Apply the Direct Comparison Test

By the Direct Comparison Test, if \( 0 \leq a_n \leq b_n \) for all \( n \) from some point onward, and \( \sum b_n \) converges, then \( \sum a_n \) converges. Here, \( a_n = \frac{1}{n! + n} \) and \( b_n = \frac{1}{n!} \). We know \( \sum_{n=1}^{\infty} \frac{1}{n!} \) converges. Since \( \frac{1}{n! + n} < \frac{1}{n!} \), the series \( \sum_{n=1}^{\infty} \frac{1}{n! + n} \) also converges.

Key concepts

Convergence of Series

When we talk about the convergence of a series, we mean that as we add more and more terms, the total sum approaches a specific value, known as the limit. Convergence is critical in calculus and mathematical analysis, as it helps to determine whether a series sums to a finite number or grows without bound.
To test for convergence, mathematicians often use comparison tests, like the Direct Comparison Test, which assists in comparing a complex series to a simpler, already known convergent series. This process helps in predicting the behavior of the series in question. In the exercise we discussed, we are tasked with determining whether the series \(\sum_{n=1}^{\infty} \frac{1}{n! + n}\) converges. By comparing it to a simpler, known convergent series \(\sum_{n=1}^{\infty} \frac{1}{n!}\), we can make informed conclusions about its convergence.

Factorial Growth

Factorial growth is a fascinating and important concept in mathematics. When we talk about a factorial, denoted by \(n!\), we mean the product of all positive integers up to \(n\). For instance, \(4! = 4 \times 3 \times 2 \times 1 = 24\).
What makes factorial growth notable is that it increases very rapidly compared to other types of growth, such as linear or even exponential growth. This rapid increase means that terms involving factorials can become very small in practical applications, as contributing terms in a series, they're likely to contribute less to the overall sum as \(n\) increases. In the exercise, this characteristic of factorials was leveraged to show that \(\frac{1}{n! + n}\) becomes very small, validating the convergence of the series.

Comparison Series

Using a comparison series is a method to determine the convergence of a complex series by comparing it to a simpler one. The comparison series should be a well-understood series whose behavior is already known. In the case we explored, the series \(\sum_{n=1}^{\infty} \frac{1}{n!}\) serves this role.
Since the factorial \(n!\) grows much faster than the linear term \(n\), the terms of the comparison series \(\frac{1}{n!}\) are easily relatable to those of the series \(\frac{1}{n! + n}\).
In the step-by-step solution, we showed that \(\frac{1}{n!} > \frac{1}{n! + n}\), which means that the series we're exploring has smaller terms than the known convergent series. This finding, validated by the Direct Comparison Test, ensures that the original series converges, bridging the complexity of new series with the known behavior of established series.

Infinite Series

An infinite series is simply the sum of an endless sequence of numbers. It might look intimidating because it involves adding up an endless list, but not all infinite series grow without stopping. Some, interestingly, come up to a final sum as result, this characteristic is known as convergence.
For example, the geometric series \(\sum_{n=0}^{\infty} \frac{1}{2^n}\) converges to 2 despite being infinite. In our exercise, dealing with the series \(\sum_{n=1}^{\infty} \frac{1}{n! + n}\), we determine convergence by ensuring the terms are sufficiently small for the total to not grow unbounded.
Infinite series are crucial in computer algorithms, signal processing, and other fields. Understanding their convergence through techniques like the Direct Comparison Test allows us to use infinite series confidently in practical applications.