Question

Test the following series for convergence or divergence. Decide for yourself which test is easiest to use, but don't forget the preliminary test. Use the facts stated above when they apply. $$\sum_{n=1}^{\infty} \frac{(n !)^{3}}{(3 n) !}$$

Short answer

The series converges by the Ratio Test.

Step-by-step solution

Preliminary Test

Check if the terms of the series \(\frac{(n !)^{3}}{(3 n)!}\) approach 0 as \(n \to \infty\). If they do not, the series diverges. Compute \(\frac{(n !)^{3}}{(3 n)!}\) for several values of \(n\) to see if the terms approach 0.

Apply the Ratio Test

Apply the Ratio Test to determine the convergence or divergence of the series. The Ratio Test states that for a series \(\sum_{n=1}^{\infty} a_n\), compute \(\rho = \lim_{n\to\infty} |a_{n+1}/a_n|\). If \(\rho < 1\), the series converges absolutely. If \(\rho > 1\), the series diverges. If \(\rho = 1\), the test is inconclusive. In our case, \(a_n = \frac{(n!)^3}{(3n)!}\).

Compute Ratio

Compute \(\frac{(n+1)^{3}}{(3(n+1))!} \div \frac{(n!)^{3}}{(3n)!} = \frac{((n+1)!)^{3}}{(3n+3)!} \cdot \frac{(3n)!}{(n!)^{3}}\) and simplify. This yields \(\rho = \lim_{n\to\infty} \frac{((n+1)!)^{3}}{(3n+3)(3n+2)(3n+1)(3n)!} \cdot \frac{(3n)!}{(n!)^{3}} = \rho = \lim_{n\to\infty} \frac{((n+1)(n!))^{3}}{(3n+3)(3n+2)(3n+1)(n!)^{3}} =\)

Simplify Further

Simplify the ratio to \(\lim_{n \to \infty} \frac{(n+1)^3(n!)^3}{(3n+3)(3n+2)(3n+1)(n!)^3} = \lim_{n \to \infty} \frac{(n+1)^3}{(3n+3)(3n+2)(3n+1)} = \lim_{n \to \infty} \frac{(n+1)^3}{27n^3}\).

Evaluate the Limit

Evaluate the limit: \(\rho = \lim_{n \to \infty} \frac{(n+1)^3}{27n^3} = \lim_{n \to \infty} \frac{n^3 \left(1 + \frac{1}{n}\right)^3}{27n^3} = \frac{1}{27}\).

Conclusion

Since \(\rho = \frac{1}{27}\), which is less than 1, the series \(\sum_{n=1}^{\infty} \frac{(n!)^3}{(3n)!}\) converges by the Ratio Test.

Key concepts

Ratio Test for Series Convergence

When testing if a series converges, the Ratio Test is a powerful tool. This test examines the limit of the ratio of successive terms in a series to determine if the series converges or diverges. For a given series \( \sum_{n=1}^{\infty} a_n \), the Ratio Test involves these steps:

• Compute the ratio \( \rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n}\right| \).
• If \( \rho < 1 \, \) the series converges absolutely.
• If \( \rho > 1 \, \) the series diverges.
• If \( \rho = 1 \), the test is inconclusive, and another test must be used.

In our example, we have Series Convergence of \( \sum_{n=1}^{\infty} \frac{(n !)^{3}}{(3 n)!} \). By applying the Ratio Test, we found \( \rho = \lim_{n \to \infty} \frac{\left( n + 1 \right)^3}{27n^3} = \frac{1}{27} \), which is less than 1, indicating that the series converges.

Preliminary Test for Series

The Preliminary Test, sometimes called the Term Test for Divergence, is a simple initial check to see if a series might diverge. The premise is straightforward:

• For the series \( \sum_{n=1}^{\infty} a_n \) to have any chance of converging, the terms \( a_n \) must approach 0 as \( n \to \infty \).
• If \( \lim_{n \to \infty} a_n \eq 0 \), the series diverges.

In our exercise, this means verifying if \( \frac{(n!)^3}{(3n)!} \to 0 \) as \( n \to \infty \). By computing the terms for increasing values of \( n \), we can see they approach 0, satisfying the preliminary check and allowing us to proceed to more detailed tests like the Ratio Test.

Understanding Factorials in Series

Factorials play an integral role in many series, especially those in combinatorics and probability. A factorial, denoted \( n! \), is the product of all positive integers up to \( n \). For example,

• \( 1! = 1 \)
• \( 2! = 2*1 = 2 \)
• \( n! = n*(n-1)*(n-2) \cdots 1 \).

In the context of series convergence, factorials can create rapidly growing terms, making some series challenging to test. For our series \( \sum_{n=1}^{\infty} \frac{(n!)^3}{(3n)!} \), the factorial terms required meticulous handling during the Ratio Test, leading us to ultimately determine that \( \lim_{n \to \infty} \frac{(n+1)^3}{27n^3} = \frac{1}{27} \). This approach helps understand how incredibly large or complex terms can impact convergence or divergence in a given series.