Question

In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series \(\sum a_{k}\) with given terms \(a_{k}\) converges, or state if the test is inconclusive. $$ a_{k}=\frac{2-4 \cdot 6 \cdots 2 k}{(2 k) !} $$

Short answer

The series converges by the Ratio Test.

Step-by-step solution

Identify the Sequence Terms

We begin by expressing the terms of the sequence given as \(a_k = \frac{2 \times 4 \times 6 \cdots 2k}{(2k)!}\). These can be seen as the product of the even numbers from \(2\) to \(2k\).

Choose the Appropriate Test

Given the factorial in the denominator, the **Ratio Test** is often more convenient because it handles factorials neatly.

Apply the Ratio Test Formula

The Ratio Test states that for the series \(\sum a_k\), if \(\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = L\), and \(L < 1\), the series converges absolutely. So, we compute \(\frac{a_{k+1}}{a_k}\).

Calculate \(a_{k+1}\)

First, express \(a_{k+1}\):\[a_{k+1} = \frac{2 \times 4 \times 6 \cdots 2k \times 2(k+1)}{(2(k+1))!}\]

Compute \(\frac{a_{k+1}}{a_k}\)

Substitute \(a_k\) and \(a_{k+1}\) into the ratio formula:\[\frac{a_{k+1}}{a_k} = \frac{(2 \times 4 \times 6 \cdots 2k \times 2(k+1))}{(2(k+1))!} \cdot \frac{(2k)!}{2 \times 4 \times 6 \cdots 2k}\]Which simplifies to:\[\frac{2(k+1)}{(2k+1)(2k+2)}\]

Evaluate the Limit

Next, find the limit:\[\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = \lim_{k \to \infty} \frac{2(k+1)}{(2k+1)(2k+2)} = \lim_{k \to \infty} \frac{2k + 2}{4k^2 + 6k + 2}\]As \(k\) becomes very large, the highest degree terms dominate:\[\lim_{k \to \infty} \frac{2}{4k} = 0\]

Conclude the Convergence

Since the limit \(L = 0 < 1\), the Ratio Test indicates that the series converges.

Key concepts

Ratio Test

The Ratio Test is a powerful tool used to determine the convergence of an infinite series, especially when factorials are present in the terms. The test is performed by examining the limit of the absolute value of the ratio of consecutive terms in the series. This is expressed as:\[L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\]If this limit \(L\) is less than 1, the series converges absolutely. If \(L\) is greater than 1, the series diverges. If \(L\) equals 1, the test is inconclusive, and other convergence tests might be needed.

• The Ratio Test is particularly handy with factorials due to the simplification that often occurs when substituting terms.
• In cases where terms involve products or powers, the Ratio Test tends to yield simpler expressions.
• It's crucial to confirm that the limit can be evaluated as \( k \) approaches infinity, focusing on the dominant terms.

Understanding the Ratio Test enhances comprehension of convergence and divergence in series, streamlining the solving process.

Factorials

Factorials, symbolized by \(!\), represent the product of an integer and all the integers below it down to 1. Factorials grow very quickly, which often makes them a major factor in determining the behavior of sequences and series.For example:- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)- More generally, \( k! = k \times (k-1) \times \, ... \, \times 2 \times 1 \)In our sequence, factorials appear in the denominator as \((2k)!\). Factorials are particularly interesting in sequences due to their rapid growth rate, which can heavily influence convergence and divergence.

• When seeing factorials in series terms, it often means the Ratio Test is a suitable choice to determine convergence.
• Simplifying expressions involving factorials requires recognizing patterns, such as cancelling out common terms in products.

Handling factorials effectively can significantly simplify the process of applying the Ratio Test or tackling other mathematical challenges.

Sequence Convergence

Sequence convergence indicates that the terms of a sequence approach a specific value or concept as they progress towards infinity. Understanding this concept is crucial in determining whether a series converges or diverges.In the context of series:

• If the limit of the sequence terms \( a_k \) as \( k \) approaches infinity is non-zero, the series is likely divergent.
• If the terms approach zero, further tests are often needed to determine actual convergence.

An example of applying these ideas is shown in finding:- The limit of \( \frac{2(k+1)}{(2k+1)(2k+2)} \) as \( k \to \infty \), which corresponds to zero, indicating convergence using the Ratio Test.Understanding sequence convergence aids in applying convergence tests effectively. Analyzing limits gives insight into the behavior of sequence terms, guiding us to determine if a series will settle at a finite value as its terms grow.