Problem 49 - Essential Calculus

Question

Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty} \frac{(-1)^{n} 2^{n}}{n !} $$

Step-by-Step Solution

Answer

The given series converges.

Step 1: Recognize the Type of Series

The given series is an alternating series, where the general term can be expressed as \(a_{n}=\frac{(-1)^{n} 2^{n}}{n !}\).

Step 2: Calculate Ratio

The ratio of consecutive terms is \(\frac{a_{n+1}}{a_{n}} = \frac{(-1)^{n+1} 2^{n+1}}{(n+1) !} ÷ \frac{(-1)^{n} 2^{n}}{n !} = \frac{(-1) 2}{n+1}\). We take the absolute value to get \(|\frac{a_{n+1}}{a_{n}}| = \frac{2}{n+1}\)

Step 3: Apply Ratio Test

According to the Ratio Test, we now calculate the limit \(\lim_{n \to \infty} \frac{2}{n+1}\). This limit is 0 which is less than 1.

Step 4: Interpreting the Result

As per the Ratio Test, if the limit of the ratio is less than 1, the series is absolutely convergent. Therefore, the given series is absolutely convergent.

Key Concepts

Convergence of series

A series is a sum of terms from a sequence of numbers. Convergence of a series occurs when the sum approaches a finite number as more terms are added. This is an essential concept in calculus and mathematical analysis.
The convergence of a series heavily depends on the behavior of its terms as more of them are included in the sum.

• A series is convergent if its terms approach zero fast enough as they extend towards infinity.
• If a series does not converge, it is said to diverge.

One of the methods used to determine if a series converges is the Ratio Test. This test involves analyzing the ratio of consecutive terms in the series. If the limit of this ratio as the terms approach infinity is less than 1, the series converges absolutely.
For our particular series, the absolute ratio of consecutive terms was determined to be 0 as it approached infinity, showing convergence.

Alternating series

An alternating series is a series whose terms alternate in sign. This means that consecutive terms switch between positive and negative. The series in our exercise, \(\sum_{n=0}^{\infty} \frac{(-1)^{n} 2^{n}}{n!}\), is a classic example. Here, the general term \(b_n = (-1)^n \frac{2^n}{n!}\) causes the sign to alternate.

• Alternating series can still converge even when positive-only series do not.
• The Alternating Series Convergence Test specifically helps determine such convergence by checking if the absolute value of terms decreases to zero.

Though we used the Ratio Test for the convergence analysis in this exercise, an alternating series might also need additional checks, especially if you suspect conditional convergence. However, as shown, the series in the exercise converges absolutely, which is a stronger form of convergence and includes the criteria for alternating series.

Absolute convergence

Absolute convergence is a strong form of convergence in series, indicating that not only does the series converge, but it also converges when all of its terms are made positive. This contrasts with conditional convergence, where the series converges in its original form, but not when it's treated with all positive terms.
Absolute convergence is important because it guarantees a few properties:

• Any rearrangement of an absolutely convergent series will still converge to the same sum.
• If a series is absolutely convergent, it remains convergent under the manipulation of terms.

In the context of the original exercise, the Ratio Test demonstrated that the alternating series converges absolutely, as the ratio limit approached zero, which is less than 1. Consequently, the series converges absolutely, meaning it satisfies convergence even if the terms are all positive, securing stronger guarantees of stability and convergence properties.