Problem 11 - Essential Calculus

Question

Determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n !} $$

Step-by-Step Solution

Answer

The given series converges.

Step 1: Write down the given series

The given series is \[ \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n !} \]

Step 2: Apply the Ratio Test

Taking the absolute value, apply the ratio test: \[ \lim_ {n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_ {n \rightarrow \infty} \left| \frac{(-1)^{n+1}/(n + 1)!}{(-1)^n/n!} \right| \]

Step 3: Simplify the Ratio

Simplify the ratio by cancelling out terms and taking the limit: \[ \lim_ {n \rightarrow \infty} \left| \frac{(-1)^{n+1}/(n + 1)!}{(-1)^n/n!} \right| = \lim_ {n \rightarrow \infty} \left| \frac{1}{n + 1} \right| = 0 \]

Step 4: Determine Convergence/Divergence

Since the limit of the ratio is less than 1, it can be concluded that the given series converges according to the Ratio Test.

Key Concepts

Ratio Test

When trying to determine the convergence of a series, one effective approach is to use the Ratio Test. The Ratio Test is particularly helpful when dealing with a series involving factorials or terms raised to a power. Here's how it works:

• Take the general term from the series, denoted as \(a_n\).
• Calculate the next term \(a_{n+1}\).
• Then, find the limit of the absolute value of the ratio of these two consecutive terms, \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).

Evaluate this limit:

• If the limit is less than 1, the series converges absolutely.
• If the limit is greater than 1 or it is infinite, the series diverges.
• If the limit is exactly 1, the test is inconclusive, and you may need to use a different method to determine convergence.

You can visualize this process as examining how each term in a series relates to its subsequent term as \(n\) becomes very large. If the terms are shrinking rapidly enough, the series will converge. In our example, the series \( \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!} \) showed a limit of 0 when passed through the Ratio Test, establishing its convergence. This makes the Ratio Test a very powerful tool for analyzing series like these.

Alternating Series

An alternating series is a series where the signs of its terms alternate between positive and negative. A typical alternating series can be written as:\[ a_1 - a_2 + a_3 - a_4 + \cdots \] or more commonly as \[ \sum_{n=0}^{\infty} (-1)^n a_n \].The series given in our example \( \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!} \)is indeed an alternating series because of the \((-1)^n\) factor, which makes each term switch signs depending on whether \(n\) is even or odd.For an alternating series, there is a specific test to determine convergence: the Alternating Series Test. According to this test:

• The series converges if the absolute value of the terms \(a_n\) is decreasing and approaches zero as \(n\) approaches infinity.

In our example, aside from using the Ratio Test, notice that the terms involve \(n!\) in the denominator. This factorial term grows very fast, which means \(\frac{1}{n!}\) gets smaller as \(n\) becomes larger, satisfying the condition of the Alternating Series Test, which confirms the convergence of the series.

Factorials

Factorials, represented with the symbol \(!\), are a mathematical operation where you multiply the number by every positive integer less than itself. For example, \(n! = n \times (n-1) \times (n-2) \times \cdots \times 1\). Factorials grow rapidly, meaning they can very quickly reach large values.

This property is significant when analyzing sequences and series.

• With factorials in the denominator, the terms of a series can decrease rapidly, aiding the convergence due to the huge growth rate of \(n!\).

In the series given: \( \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!} \), notice how \(n!\) in the denominator quickly dominates the series terms, driving them towards zero as \(n\) increases.

Because factorials grow so quickly, they are often used in various mathematical expressions that require the terms to diminish swiftly and ensure convergence.
This is why expressions involving factorials often appear in problems involving convergence, such as power series or exponential functions.