Question

Use the ratio test to find whether the following series converge or diverge: $$\sum_{n=0}^{\infty} \frac{5^{n}(n !)^{2}}{(2 n) !}$$

Short answer

The series diverges.

Step-by-step solution

- Write down the ratio test formula

To use the ratio test, consider the series \ \( \sum_{n=0}^{\infty} a_n \). The ratio test involves computing \ \(\lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| \). If the limit is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit equals 1, the test is inconclusive.

- Express \(a_n\) for the given series

For the given series, identify \ \(a_n = \frac{5^n (n!)^2}{(2n)!} \).

- Find \(a_{n+1}\)

Substitute \ \(n+1\) in place of \ \(n\) in the expression for \ \(a_n\), obtaining \ \(a_{n+1} = \frac{5^{n+1} ((n+1)!)^2}{(2(n+1))!} = \frac{5^{n+1} ((n+1)!)^2}{(2n+2)!} \).

- Form the ratio \(\frac{a_{n+1}}{a_n}\)

Compute the ratio: \ \[\frac{a_{n+1}}{a_n} = \frac{5^{n+1} ((n+1)!)^2}{(2n+2)!} \cdot \frac{(2n)!}{5^n (n!)^2} = 5 \cdot \frac{(n+1)^2}{(2n+2)(2n+1)}.\] Simplify the ratio to: \ \[\frac{a_{n+1}}{a_n} = 5 \cdot \frac{n^2 + 2n + 1}{4n^2 + 6n + 2}.\]

- Take the limit as \(n\) approaches infinity

Evaluate the limit: \ \[\lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{{n \to \infty}} 5 \cdot \frac{n^2 + 2n + 1}{4n^2 + 6n + 2} = 5 \cdot \frac{1}{4} = \frac{5}{4}.\]

- Conclude the test

Since the limit \(\frac{5}{4} > 1\), by the ratio test, the given series \(\sum_{n=0}^{\infty} \frac{5^n (n!)^2}{(2n)!}\) diverges.

Key concepts

infinite series

An infinite series is the sum of the terms of an infinite sequence. Imagine you have a sequence of numbers: a1, a2, a3, and so on, continuing without end. When you add these numbers together, you get an infinite series: a1 + a2 + a3 +.... These series can behave in different ways. Sometimes, the sum reaches a finite number even though there are infinitely many terms. Other times, the series grows without bound.

convergence and divergence

In the context of infinite series, convergence and divergence are crucial concepts. When we say a series converges, we mean the sum of the series approaches a specific, finite number as more terms are added. Conversely, if a series diverges, the sum either goes to infinity, negative infinity, or does not approach any specific number as terms are added. The behavior of a series (converging or diverging) helps us understand if the sum results in a meaningful number or not.

limit comparison test

The limit comparison test is a method used to determine whether an infinite series converges or diverges. It is particularly useful when dealing with series that are tough to handle directly. This test compares the given series to a known benchmark series with well-understood convergence or divergence behavior. Given two positive term series \textteam.tex{\frac{\textlambda \(\frac{a_n}{b_n}}\)}, where the given series (`a_n`) is compared against a known series (`b_n`). If the ratio of \frac{\textlambda \(\frac{a_n}{b_n}}\). If this ratio is a positive finite number, both series either converge or both diverge. This technique can significantly simplify the process of determining a series' behavior.