Problem 15 - Single Variable Calculus

Question

Determine whether the series is convergent, absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{2^{n}}{n ! n}\)

Step-by-Step Solution

Answer

The series \(\sum_{n=1}^{\infty} \frac{2^{n}}{n!n}\) is absolutely convergent, as determined by applying the Ratio Test and finding that the limit \(L = \lim_{n \to \infty} \frac{2}{n+1} = 0\) which is less than 1.

Step 1: Find the Absolute Value of the Terms in the Series

(ComputerAssistantEnteringMathMode) Evaluate the absolute value of the terms in the series: \(|a_n| = \left|\frac{2^{n}}{n!n}\right| = \frac{2^{n}}{n!n}\) (ComputerAssistantExitingMathMode) Since the terms are already positive, the absolute value does not change the series.

Step 2: Apply the Ratio Test

(ComputerAssistantEnteringMathMode) Apply the Ratio Test to determine the absolute convergence of the series: \(L = \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{\frac{2^{n+1}}{(n+1)!(n+1)}}{\frac{2^n}{n!n}}\) (ComputerAssistantExitingMathMode)

Step 3: Simplify and Evaluate the Limit

(ComputerAssistantEnteringMathMode) Simplify the limit expression: \(L = \lim_{n \to \infty} \frac{2^{n+1}n!n}{2^n(n+1)!(n+1)} = \lim_{n \to \infty} \frac{2n!}{(n+1)(n+1)!}\) (ComputerAssistantExitingMathMode) Now, evaluate the limit: \(L = \lim_{n \to \infty} \frac{2n!}{(n+1)(n+1)!} = \lim_{n \to \infty} \frac{2}{n+1} = 0\) Since \(L < 1\), the series is absolutely convergent. Since the series is absolutely convergent, we don't need to test for conditional convergence or divergence.

Step 4: Conclusion

The given series \(\sum_{n=1}^{\infty} \frac{2^{n}}{n!n}\) is absolutely convergent.

Key Concepts

Absolutely Convergent

A series is considered absolutely convergent when the series formed by taking the absolute value of its terms is convergent. In simple terms, if you take each term of the series and ignore whether it is positive or negative, and the resulting series converges, then the original series is absolutely convergent.
This concept is crucial because it guarantees convergence. Unlike other forms of convergence, absolute convergence implies that the series will still converge even if each of its components is positive. This happens due to its strong criteria which ensures the series sums up towards a finite limit.
For the given exercise, we found that \( \sum_{n=1}^{\infty} \frac{2^{n}}{n!n} \) is absolutely convergent as the absolute value also converges.

Ratio Test

The ratio test is a popular method used to determine the convergence of an infinite series. It involves taking the limit of the absolute value of the ratio of consecutive terms.
To apply this test, we calculate:

• \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \)

If \( L < 1 \), the series is absolutely convergent. If \( L > 1 \), the series is divergent. If \( L = 1 \), the test is inconclusive, and other methods must be used.
In our example, the limit \( L \) evaluated to 0, which is less than 1, confirming the absolute convergence of the series. This effective rule helps quickly assess series by examining the behavior of their ratios.

Factorial

The factorial, represented by an exclamation mark \( n! \), is the product of all positive integers up to \( n \). For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials grow very quickly, which often helps in evaluating the convergence of series.
In our exercise, the term \( n! \) significantly contributed to the convergence of the series. Factorials tend to outgrow exponential terms like \( 2^n \) rapidly, often resulting in the terms of the series diminishing towards zero, thus aiding convergence.
Understanding the impact of factorials in a series helps to predict whether the series terms approach zero fast enough to meet convergence criteria.

Infinite Series

An infinite series is a sum of infinitely many terms, expressed in the form: \( \sum_{n=1}^{\infty} a_n \). Exploring infinite series involves understanding whether the sequence of its partial sums converges to a limit.
To determine this, various tests and criteria are applied, such as the ratio test used in our exercise.

• A series is convergent if these partial sums approach a specific value.
• If they do not approach any limit, the series is divergent.

Infinite series can represent complex functions or constants in mathematical analysis. As seen in our problem, series sometimes converge because their terms shrink rapidly enough, a property frequently guaranteed by factorial terms in the denominator.