Question

In the following exercises, use an appropriate test to determine whether the series converges. $$ a_{k}=1 /\left(\begin{array}{l} 2 k \\ k \end{array}\right) $$

Short answer

The series converges by the ratio test.

Step-by-step solution

Identify the Series

The given series has terms of the form \( a_k = \frac{1}{\binom{2k}{k}} \). We need to determine whether this series \( \sum_{k=1}^{\infty} a_k \) converges.

Understand the Binomial Coefficient

The binomial coefficient \( \binom{2k}{k} \) is calculated using the formula \( \binom{2k}{k} = \frac{(2k)!}{(k!)^2} \). This represents the number of ways to choose \( k \) elements from \( 2k \) elements.

Use Asymptotic Approximation

For large \( k \), the binomial coefficient \( \binom{2k}{k} \) is approximately \( \frac{4^k}{\sqrt{\pi k}} \). Therefore, \( a_k \approx \frac{1}{\frac{4^k}{\sqrt{\pi k}}} = \frac{\sqrt{\pi k}}{4^k} \).

Apply the Ratio Test

To determine convergence, apply the ratio test. Consider the ratio \( \frac{a_{k+1}}{a_k} \). Calculate it as:\[\frac{a_{k+1}}{a_k} = \frac{\sqrt{\pi(k+1)}}{4^{k+1}} \times \frac{4^k}{\sqrt{\pi k}} = \frac{\sqrt{\pi(k+1)}}{4\sqrt{\pi k}} = \frac{\sqrt{k+1}}{4\sqrt{k}} \].

Simplify the Ratio

Simplify \( \frac{\sqrt{k+1}}{4\sqrt{k}} \) to check its limit as \( k \to \infty \):\[\lim_{k \to \infty} \frac{\sqrt{k+1}}{4\sqrt{k}} = \lim_{k \to \infty} \frac{\sqrt{k}\sqrt{1 + \frac{1}{k}}}{4\sqrt{k}} = \lim_{k \to \infty} \frac{1}{4}\sqrt{1+\frac{1}{k}} = \frac{1}{4}.\]

Convergence Conclusion

Since \( \lim_{k \to \infty} \frac{\sqrt{k+1}}{4\sqrt{k}} = \frac{1}{4} < 1 \), the ratio test confirms that the series \( \sum_{k=1}^{\infty} a_k \) converges.

Key concepts

Ratio Test

The Ratio Test is a powerful tool used to determine whether an infinite series converges. To apply this test, you analyze the ratio of successive terms of a series. If the limit of the absolute value of the ratio of consecutive terms is less than one, the series converges. Start by considering a series in the form \( \sum_{k=1}^{\infty} a_k \). Calculate the ratio \( \frac{a_{k+1}}{a_k} \), and evaluate \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \). If this limit is less than 1, the series converges absolutely. If it is greater than 1, or if it is infinite, the series diverges. If it equals 1, the test is inconclusive.

In our exercise, the ratio \( \frac{a_{k+1}}{a_k} = \frac{\sqrt{k+1}}{4\sqrt{k}} \) was calculated, and the final limit turned out to be \( \frac{1}{4} \). Since \( \frac{1}{4} < 1 \), this indicates convergence of the series \( \sum_{k=1}^{\infty} \frac{1}{\binom{2k}{k}} \). This example showcases the effectiveness of the Ratio Test in confirming convergence.

Binomial Coefficient

The binomial coefficient, denoted as \( \binom{n}{k} \), represents the number of ways to select \( k \) elements from a set of \( n \) elements without considering the order. In mathematics, it is expressed as:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

where \( n! \) ("n factorial") is the product of all positive integers up to \( n \).

In the exercise, the series uses \( \binom{2k}{k} \), calculated using \( \binom{2k}{k} = \frac{(2k)!}{(k!)^2} \). This specific binomial coefficient counts the number of ways to choose equal subsets from a larger set, a typical application in combinatorics.
Understanding the behavior and calculation of binomial coefficients allows us to simplify series expressions, as seen in the exercise where further analysis leads to applying an asymptotic approximation.

Asymptotic Approximation

Asymptotic approximation helps simplify complex mathematical expressions for large values of variables. When dealing with binomial coefficients such as \( \binom{2k}{k} \), exact calculations can be cumbersome, especially for large \( k \). Asymptotic approximations allow us to estimate these values more feasibly.

In the given problem, we applied the asymptotic approximation:

• \( \binom{2k}{k} \approx \frac{4^k}{\sqrt{\pi k}} \) for large \( k \)

This approximation significantly simplifies the term \( a_k = \frac{1}{\binom{2k}{k}} \) to \( a_k \approx \frac{\sqrt{\pi k}}{4^k} \). This expression unveils the dominant factors in the sequence, enabling us to apply further methods such as the Ratio Test to determine convergence. Asymptotic approximation thus serves as a practical strategy in complex series analysis.