Problem 5 - Thomas Calculus

Question

In Exercises \(1-8,\) use the Ratio Test to determine if each series converges absolutely or diverges. $$ \sum_{n=1}^{\infty} \frac{n^{4}}{(-4)^{n}} $$

Step-by-Step Solution

Answer

The series converges absolutely by the Ratio Test.

Step 1: Identify the General Term

For the given series \( \sum_{n=1}^{\infty} \frac{n^{4}}{(-4)^{n}} \), identify the general term \( a_n = \frac{n^{4}}{(-4)^{n}} \).

Step 2: Apply the Ratio Test Formula

The Ratio Test involves computing \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). Substitute \( a_n = \frac{n^{4}}{(-4)^n} \) into this formula: \( \lim_{n \to \infty} \left| \frac{\frac{(n+1)^4}{(-4)^{n+1}}}{\frac{n^4}{(-4)^n}} \right| \).

Step 3: Simplify the Ratio Expression

Simplify the ratio \( \frac{a_{n+1}}{a_n} \) expression: \[ \left| \frac{(n+1)^4 (-4)^n}{n^4 (-4)^{n+1}} \right| = \left| \frac{(n+1)^4}{n^4} \cdot \frac{1}{4} \right| \].

Step 4: Evaluate the Limit

Evaluate the limit \( \lim_{n \to \infty} \left| \frac{(n+1)^4}{n^4} \cdot \frac{1}{4} \right| \). Focus first on \( \lim_{n \to \infty} \frac{(n+1)^4}{n^4} \): Factor out \( n^4 \) from the numerator to get \( \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^4 = 1^4 = 1 \). Therefore, the overall limit is \( 1 \times \frac{1}{4} = \frac{1}{4} \).

Step 5: Conclusion from the Ratio Test

According to the Ratio Test, if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \), the series converges absolutely. Since \( \frac{1}{4} < 1 \), the series \( \sum_{n=1}^{\infty} \frac{n^{4}}{(-4)^{n}} \) converges absolutely.

Key Concepts

Convergence and Divergence of Series

In calculus, when we work with series, an important question is whether the series converges or diverges. In simple terms, convergence means that the sum of all terms in the series approaches a specific value as the number of terms increases to infinity. Divergence, on the other hand, means the sum does not approach a particular value, often leading to infinity or oscillating without settling down.

We use different tests to determine the convergence or divergence of a series. One such method is the Ratio Test, which is particularly useful for series with factorials or exponential terms. By examining the behavior of the ratio of consecutive terms, we can decide whether a series converges absolutely, converges conditionally, or diverges.

Keep in mind:

• If the limit of the ratio is less than 1, the series converges absolutely.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the Ratio Test is inconclusive.

General Term of a Series

For any series, the general term plays a crucial role in analyzing its behavior. The general term is essentially an expression that describes every term in the series. It serves as the building block for understanding the whole sequence.

In the series presented, the general term is given by:
\[a_n = \frac{n^4}{(-4)^n}\]
This formula shows that each term in the series is a fraction with \(n^4\) on top and \((-4)^n\) on the bottom. Knowing this general term is the first step in applying tests like the Ratio Test as it helps calculate the ratio of consecutive terms.

Limit Evaluation in Calculus

The concept of limits in calculus is vital for determining the behavior of functions and sequences as they approach a certain point. When using the Ratio Test for series, it’s essential to evaluate the limit of the ratio of successive terms as \(n\) approaches infinity.

In our example, you'll see that after substituting the general terms into the ratio formula, we simplify to:
\[\lim_{n \to \infty} \left| \frac{(n+1)^4}{n^4} \cdot \frac{1}{4} \right|\]
Here, the expression can be broken into the product of two limits. By simplifying \(\left(1 + \frac{1}{n}\right)^4\) and factoring out \(n^4\), we evaluate that it approaches 1. Therefore, the entire limit evaluates to \(\frac{1}{4}\). This assessment is crucial because a limit result less than one indicates absolute convergence when using the Ratio Test.

Absolute Convergence

Absolute convergence is a strong form of convergence for series. A series is said to converge absolutely if the series formed by taking the absolute values of each term also converges. This ensures that the series converges regardless of the arrangement of its terms.

When using the Ratio Test, if the calculated limit of the ratio of successive terms is less than 1, the series converges absolutely. In the given series, we've determined that the limit is \(\frac{1}{4}\), which is less than 1. This conclusion tells us that the series not only converges but does so absolutely, which gives more information about its behavior compared to merely checking for conditional convergence.