Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine convergence or divergence for each of the series. Indicate the test you use. $$ \sum_{n=1}^{\infty} \frac{4^{n}+n}{n !} $$

Step-by-Step Solution

Answer

The series converges by the Ratio Test.

Step 1: Recognize the Form

This series has the form \( \sum_{n=1}^{\infty} \frac{a_n}{b_n} \), where \( a_n = 4^{n} + n \) and \( b_n = n! \). The factorial in the denominator suggests this might be a good candidate for the Ratio Test.

Step 2: Set Up the Ratio Test

The Ratio Test involves considering the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1} / b_{n+1}}{a_n / b_n} \right| \). For our series, this is expressed as \( \lim_{n \to \infty} \left| \frac{4^{n+1} + (n+1)}{(n+1)!} \cdot \frac{n!}{4^n + n} \right| \).

Step 3: Simplify the Expression

Simplify the ratio expression using algebra: \( \lim_{n \to \infty} \frac{4^{n+1} + n+1}{(n+1)(4^n + n)} \). Note that \( 4^{n+1} = 4 \cdot 4^n \). Thus, the expression becomes \( \lim_{n \to \infty} \frac{4 \cdot 4^n + n + 1}{(n+1) \cdot 4^n + n(n+1)} \).

Step 4: Analyze Dominant Terms

As \( n \) becomes large, the term \( 4^{n+1} = 4 \cdot 4^n \) dominates in both the numerator and the denominator. The limit simplifies to the ratio of their coefficients: \( \lim_{n \to \infty} \frac{4 \cdot 4^n}{n \cdot 4^n} = \frac{4}{n} \).

Step 5: Evaluate the Limit

Evaluate \( \lim_{n \to \infty} \frac{4}{n} = 0 \). Since 0 is less than 1, according to the Ratio Test, our series \( \sum_{n=1}^{\infty} \frac{4^{n}+n}{n!} \) converges.

Key Concepts

Ratio Test

The Ratio Test is an effective technique for determining the convergence or divergence of an infinite series. It is especially useful when dealing with series that involve factorials or exponential terms. The basic idea of the Ratio Test is to analyze the behavior of the series' terms as they progress to infinity.

Here’s how you apply the Ratio Test in a step-by-step manner:

• For a series \( \sum_{n=1}^{\infty} a_n \), determine the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).
• If \( L < 1 \), the series absolutely converges.
• If \( L > 1 \) or \( L \) is infinite, the series diverges.
• If \( L = 1 \), the test is inconclusive, and other methods must be used.

In our specific exercise, the series \( \sum_{n=1}^{\infty} \frac{4^{n}+n}{n!} \) makes it a perfect candidate for the Ratio Test due to the factorial \( n! \) and exponential \( 4^n \) present in the terms.

Factorial Series

A factorial series is a sequence where the terms are divided by factorial numbers. This kind of series often appears in calculus and can be tackled efficiently using the Ratio Test. Factorials (denoted as \( n! \)) are a product of all positive integers up to \( n \). As \( n \) grows, \( n! \) increases very rapidly.

Factorial series, such as \( \sum \frac{a_n}{n!} \), typically converge because the factorial in the denominator grows much faster than polynomial or even exponential terms in the numerator.

In our example,

• The denominator \( n! \) suggests dominance over any linear or exponential terms in the numerator.
• This makes the series a likely candidate for convergence, which the Ratio Test confirms analytically.

Thus, incorporating factorial terms in a series naturally disposes it to possess convergent properties, especially when tested through convergence tests.

Algebraic Simplification

Algebraic simplification is a crucial step in applying the Ratio Test, especially for series with complex terms. Simplifying the expressions allows us to find limits more accurately.

In our series, algebraic simplification helps to reduce the fraction complexity:

• We rewrite \( 4^{n+1} \) as \( 4 \cdot 4^n \).
• This step makes it easier to identify the dominant terms in the expression.
• Clearing out terms that diminish quickly as \( n \to \infty \) helps in focusing on the significant parts of the series.

Through careful simplification, we observe that the leading term \( 4 \cdot 4^n \) both in the numerator and the denominator simplifies our limit to \( \frac{4}{n} \), which reduces to zero as \( n \) becomes very large. This simplification is key to establishing the convergence of the series quickly and effectively.

Convergence Tests

Convergence tests are essential tools for understanding the behavior of series. Beyond the Ratio Test, there are several other notable tests, each suitable for specific types of series.

Some commonly used convergence tests include:

• Root Test: Similar to the Ratio Test, but instead uses roots to analyze term behavior.
• Comparison Test: Compares the series with a known benchmark series for convergence or divergence.
• Integral Test: Involves integrating a function related to the terms of the series to determine convergence.
• Alternating Series Test: Specifically for series with terms that alternate in sign.

In mathematical analysis, choosing the correct test such as the Ratio Test can greatly simplify solving problems about series convergence. It provides a structured pathway, allowing mathematicians to make precise conclusions about the infinite nature of series like \( \sum_{n=1}^{\infty} \frac{4^{n}+n}{n!} \). Understanding each test thoroughly equips students with a robust set of tools for tackling series inquiries.