Question

Use the ratio test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is given in the following problems. State if the ratio test is inconclusive. $$ \sum_{n=1}^{\infty} \frac{(2 n) !}{(2 n)^{n}} $$

Short answer

The series converges by the ratio test.

Step-by-step solution

Ratio Test Formula

The ratio test involves finding the limit of the absolute value of the ratio of consecutive terms of a series. For the series \( \sum a_n \), the ratio \( r \) is given by \( r = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). If \( r < 1 \), the series converges; if \( r > 1 \), it diverges; and if \( r = 1 \), the test is inconclusive.

Define Terms

We define \( a_n = \frac{(2n)!}{(2n)^n} \). Therefore, \( a_{n+1} = \frac{(2(n+1))!}{(2(n+1))^{n+1}} \).

Compute \( \frac{a_{n+1}}{a_n} \)

To apply the ratio test, compute the ratio of \( a_{n+1} \) to \( a_n \): \[ \frac{a_{n+1}}{a_n} = \frac{(2(n+1))!}{(2(n+1))^{n+1}} \times \frac{(2n)^n}{(2n)!} \].

Simplify the Ratio

Substitute \((2(n+1))! = (2n+2)(2n+1)(2n)!\) into the equation: \[ \frac{(2n+2)(2n+1)(2n)!}{(2n+2)^{n+1} (2n)!} \times \frac{(2n)^n}{(2n)!} \]. Cancel \((2n)!\): \[ \frac{(2n+2)(2n+1)(2n)^n}{(2n+2)^{n+1}} \].

Simplify Further

Expand the expression: \[ \frac{(2n+2)(2n+1)}{(2n+2)^{n+1}} = \frac{(2n+2)(2n+1)}{(2n+2)(2n+2)^n} \]. Cancelling \(2n+2\) gives: \[ \frac{(2n+1)}{(2n+2)^n} \]. Further simplification as \(n \to \infty\).

Analyze Limit

Consider the limit: \[ \lim_{n \to \infty} \left| \frac{(2n+1)}{(2n+2)^n} \right| \]. Notice the exponential function in the denominator grows faster than the linear function in the numerator. Therefore, \( \lim_{n \to \infty} \left| \frac{1}{(2n+2)^n} \right| = 0 \).

Conclusion of Ratio Test

Since \( r = 0 < 1 \), by the ratio test, the series \( \sum_{n=1}^{\infty} \frac{(2n)!}{(2n)^n} \) converges.

Key concepts

Convergence of Series

When we talk about the **convergence of a series**, we are describing whether the sum of an infinite sequence of terms results in a finite number. To check for convergence, we can use various tests, and the Ratio Test is one of them. This test examines whether the ratio of consecutive terms in the sequence results in a limit that is less than, more than, or equal to one. Here’s a quick guide:

• If the ratio limit you get is less than one, the series definitely converges.
• If it's more than one, the series diverges.
• If it's exactly one, then the Ratio Test doesn't give us a clear answer, meaning it's inconclusive.

By applying these principles, we can effectively determine if a series converges from the behavior of its terms when they grow extremely large.

Factorials in Series

**Factorials** are a way to represent the product of all positive integers up to a given number, denoted as \( n! \). They frequently appear in mathematical series, particularly in problems involving permutations, combinations, or power series. When included in a series, they can greatly influence the behavior of the series because factorials grow very rapidly with increasing values of \( n \).

• In many series, factorials can cause terms to grow large, potentially leading the series to diverge unless counterbalanced by rapidly decreasing terms.
• In our problem, \((2n)!\) plays a pivotal role, as the rapid growth of the factorial is tempered by an exponential term in the denominator.
• This creates a balance that must be carefully analyzed, typically requiring advanced techniques like the Ratio Test to determine the series' convergence.

Limit of Sequences

The **limit of sequences** is a fundamental concept when analyzing series. The limit tells us what value (if any) a sequence approaches as the number of terms goes to infinity.When applying the Ratio Test, it’s important to calculate the limit of the ratio of consecutive terms. In our case, we found:\[\lim_{n \to \infty} \left| \frac{(2n+1)}{(2n+2)^n} \right|\]Here, we observe that the exponential term in the denominator grows faster than the linear term in the numerator as \(n\) becomes very large. This results in the entire fraction approaching zero.
Such behavior indicates that the terms of the series decrease rapidly, leading to the conclusion, through the Ratio Test, that the series converges. This outcome showcases how understanding the limits of sequences can shed light on the nature of a series, helping to predict its convergence or divergence.