Problem 1 - Thomas Calculus

Question

In Exercises \(1-8,\) use the Ratio Test to determine if each series converges absolutely or diverges. $$\sum_{n=1}^{\infty} \frac{2^{n}}{n !}$$

Step-by-Step Solution

Answer

The series converges absolutely.

Step 1: Define the general term of the series

The given series is \(\sum_{n=1}^{\infty} \frac{2^{n}}{n !}\). The general term \(a_n\) of the series is \(a_n = \frac{2^n}{n!}\).

Step 2: Apply the Ratio Test

To apply the Ratio Test, consider \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\). If \(L < 1\), the series converges absolutely. If \(L > 1\), the series diverges. If \(L = 1\), the test is inconclusive.

Step 3: Calculate the ratio \( \frac{a_{n+1}}{a_n} \)

Compute the next term \(a_{n+1} = \frac{2^{n+1}}{(n+1)!}\). Now find \(\frac{a_{n+1}}{a_n} = \frac{2^{n+1}}{(n+1)!} \cdot \frac{n!}{2^n} = \frac{2 \cdot 2^n \cdot n!}{2^n \cdot (n+1) \cdot n!} = \frac{2}{n+1}\).

Step 4: Take the limit as \(n\) approaches infinity

Now, calculate \(L = \lim_{n \to \infty} \left| \frac{2}{n+1} \right| = \lim_{n \to \infty} \frac{2}{n+1} = 0\).

Step 5: Conclusion of the Ratio Test

Since \(L = 0 < 1\), the series \(\sum_{n=1}^{\infty} \frac{2^n}{n!}\) converges absolutely.

Key Concepts

Convergence of Series

The concept of convergence is crucial in understanding series. When we talk about the convergence of a series, we are interested in understanding whether the sum of its infinite terms approaches a specific value as we continue to add terms.
\[ \sum_{n=1}^{\infty} a_n \]
Key things to remember about convergence include:

• If the series converges, it means the sequence of partial sums has a finite limit.
• The series diverges if it does not converge, meaning the sum either increases without bound or oscillates.
• The Ratio Test is a popular method for testing convergence, especially for series with factorials or exponential components.

Understanding convergence gives insight into the behavior of the series overall and helps in determining the value or approximation of the series, if it converges.

Factorial in Series

Factorials often occur in series, particularly in problems involving permutations, combinations, or any series approximating exponential functions. A factorial, represented by \( n! \), is a product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

In terms of series representation:

• Factorials grow very quickly as \( n \) increases, which effectively reduces the size of subsequent terms in a denominator.


• This characteristic is useful in determining convergence, as factorials in the denominator often lead to a series that converges.
• Knowing how factorials interact within a series can simplify calculations using tests like the Ratio Test.

This insight helps when applying convergence tests, as series like our original example often become manageable due to these rapidly growing terms in the denominator.

Limit of a Sequence

The limit of a sequence is a foundational concept in calculus and analysis. It tells us about the behavior of a sequence as the index grows indefinitely. In our context with series, understanding limits is essential for applying the Ratio Test.

Here’s what to know:

• A sequence \( \{a_n\} \) converges to a limit \( L \) if, as \( n \to \infty \), \( a_n \) approaches \( L \).


• Determining the limit of the ratio \( \frac{a_{n+1}}{a_n} \) is crucial in the Ratio Test.
• If the limit \( L < 1 \), the original series converges absolutely. If \( L > 1 \), it diverges. If \( L = 1 \), the test is inconclusive.

In our exercise, the limit evaluated to 0, which confirmed the absolute convergence of the series. Using limits helps us effectively apply tests and understand series behavior beyond basic calculations.