Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that each series converges absolutely. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{2^{n}}{n !} $$

Step-by-Step Solution

Answer

The series converges absolutely by the ratio test.

Step 1: Identify the Series

The given series is \( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{2^n}{n!} \). This is an alternating series because the terms involve \((-1)^{n+1}\).

Step 2: Check for Absolute Convergence

For absolute convergence, consider the series with absolute values: \( \sum_{n=1}^{\infty} \left| (-1)^{n+1} \frac{2^n}{n!} \right| = \sum_{n=1}^{\infty} \frac{2^n}{n!} \). This is a non-alternating series.

Step 3: Apply the Ratio Test

To determine absolute convergence, apply the ratio test to \( \sum_{n=1}^{\infty} \frac{2^n}{n!} \). Calculate the limit: \( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} \) where \( a_n = \frac{2^n}{n!} \).

Step 4: Calculate the Ratio of Consecutive Terms

Find \( a_{n+1} = \frac{2^{n+1}}{(n+1)!} \) and \( a_n = \frac{2^n}{n!} \). Thus, the ratio is \( \frac{a_{n+1}}{a_n} = \frac{2^{n+1}}{(n+1)!} \cdot \frac{n!}{2^n} = \frac{2}{n+1} \).

Step 5: Evaluate the Limit

Evaluate \( \lim_{n \to \infty} \frac{2}{n+1} = 0 \). Since the limit is 0 (less than 1), the series converges absolutely by the ratio test.

Key Concepts

Alternating Series

An alternating series is defined by the terms that switch signs each time a new term is added. This alternation is typically due to factors like \((-1)^n\) or \((-1)^{n+1}\), creating a pattern of positivity and negativity in consecutive terms. Alternating series often look like this: (a) ... + (-1)^n a_n + (-1)^{n+1} a_{n+1} ... These types of series are important in the world of mathematical analysis because they have specific convergence properties. One of the key properties is known as the **Alternating Series Test**, which provides criteria under which such a series converges. A series \(\sum (-1)^n a_n\) will converge if:

• The sequence \(a_n\) is decreasing: \(a_{n+1} \leq a_n\) for all \(n\).
• The limit of \(a_n\) as \(n\) goes to infinity is zero: \(\lim\limits_{n \to \infty} a_n = 0\).

However, our task here was to determine absolute convergence, which goes beyond just regular convergence.

Ratio Test

The Ratio Test is a tool we apply to determine whether an infinite series is absolutely convergent, conditionally convergent, or divergent. It involves analyzing the limit of the ratio of successive terms. Consider a series \(\sum a_n\), the Ratio Test examines the limit:\[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \]Here's how the test works:

• If \(L < 1\), the series absolutely converges.
• If \(L > 1\) or \(L\) is infinite, the series diverges.
• If \(L = 1\), the test is inconclusive.

This technique is especially useful for series involving factorials or exponential terms, as the simplification of these terms often becomes straightforward when using ratios. In our scenario, by finding \(L = 0\), it was clear that the series converges absolutely, meaning that series with positive terms sum up to a finite number.

Convergent Series

A convergent series in mathematics is one whose terms add up to a finite limit as the number of terms grows indefinitely. This contrasts sharply with divergent series, for which the sums grow without bound. Convergent series can be understood through different "tests" or checks, one of which is the Ratio Test discussed earlier.For convergence, we seek a sequence \(\sum a_n\) where the partial sums approach a particular limit as more terms are added:\[ \lim_{n \to \infty} S_n = S \]Here, \(S\) is finite. Convergence appears in various contexts, sometimes more naturally when tested for absolute convergence. After confirming absolute convergence using the ratio test, there is no need to apply other convergent tests, as absolute convergence guarantees convergence.

Factorial

Factorials, denoted by \(n!\), represent the product of all positive integers up to \(n\). They play a critical role in series evaluation, especially with the alternating and ratio test environments.For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).Factorials grow extremely large quickly, which makes them useful in terms where large numbers dampen the growing effect of other expressions, such as exponential terms.In our exercise, each term of the series involves a factorial, \(n!\), in the denominator:\[ \frac{2^n}{n!} \]This formulation helps the series terms "shrink" fast enough for convergence, because while \(2^n\) grows exponentially, \(n!\) increases even faster.Factorials are often used in probability, combinatorics, and series to represent permutations and expand into infinite series solutions.