Problem 22 - Calculus: Early Transcendentals

Question

Use the Ratio Test to determine whether the series is convergent or divergent. \( \frac {2}{3} + \frac {2 \cdot 5}{3 \cdot 5} + \frac {2 \cdot 5 \cdot 8}{3 \cdot 5 \cdot 7} + \frac {2 \cdot 5 \cdot 8 \cdot 11}{3 \cdot 5 \cdot 7 \cdot 9} + \cdot \cdot \cdot \)

Step-by-Step Solution

Answer

The Ratio Test is inconclusive for this series.

Step 1: Identify the General Term

Let's write the general term of the given series. Denote the term as \( a_n \). The provided series appears to follow the pattern:\[a_n = \frac{2 \cdot 5 \cdot (3n - 1)}{3 \cdot 5 \cdot (2n + 1)}\]where consecutive terms include products increasing by 3 for the numerator and 2 for the denominator.

Step 2: Apply the Ratio Test Formula

The Ratio Test states that a series \( \sum a_n \) converges if the limit\[ \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| < 1 \]Diverges if\[ \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| > 1 \]And is inconclusive if\[ \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| = 1 \]

Step 3: Find the (n+1)-th Term

For the \((n+1)\)-th term, substitute \(n+1\) as follows:\[a_{n+1} = \frac{2 \cdot 5 \cdot (3(n+1) - 1)}{3 \cdot 5 \cdot (2(n+1) + 1)} = \frac{2 \cdot 5 \cdot (3n + 2)}{3 \cdot 5 \cdot (2n + 3)}\]

Step 4: Compute the Ratio

Plug in the terms into the Ratio Test expression:\[\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{\frac{2 \cdot 5 \cdot (3n + 2)}{3 \cdot 5 \cdot (2n + 3)}}{\frac{2 \cdot 5 \cdot (3n - 1)}{3 \cdot 5 \cdot (2n + 1)}} \right| = \frac{(3n + 2)(2n + 1)}{(3n - 1)(2n + 3)}\]

Step 5: Simplify the Expression

Simplify the expression from the previous step:\[\frac{(3n + 2)(2n + 1)}{(3n - 1)(2n + 3)}\]Expand and perform algebraic simplification, leading to:\[\lim_{{n \to \infty}} \frac{6n^2 + 3n + 4n + 2}{6n^2 + 9n - 2n - 3} = \lim_{{n \to \infty}} \frac{6n^2 + 7n + 2}{6n^2 + 7n - 3}\]

Step 6: Evaluate the Limit

To find the limit, divide each term by \(n^2\):\[\lim_{{n \to \infty}} \frac{6 + \frac{7}{n} + \frac{2}{n^2}}{6 + \frac{7}{n} - \frac{3}{n^2}} = \frac{6}{6} = 1\]The limit of the ratio is 1.

Step 7: Conclude Using the Ratio Test

Since the limit derived from the Ratio Test is 1, the Ratio Test is inconclusive for this series.

Key Concepts

Convergence of Series

Convergence of a series indicates that as we keep adding up the terms, the sum approaches a finite number. Take an infinite series, which is just a collection of terms added together without end. When we say a series converges, it means that even if we keep adding terms, the total sum gets closer and closer to a specific value.
This idea is pivotal in understanding series because it tells us whether our series "settles down" to a specific number.

To determine convergence, we use the **Ratio Test**.
The Ratio Test provides a rule of thumb: calculate the ratio of the absolute value of consecutive terms. Specifically, we look at the limit of the ratio of \(\left| \frac{a_{n+1}}{a_n} \right|\) as \( n \) approaches infinity.
If this limit is less than 1, the series converges.

In our exercise, upon simplifying and finding the limit of the ratio, we achieved a result of 1, which makes the test inconclusive for convergence.

Divergence of Series

On the flip side of convergence is divergence, where the sum of the series keeps increasing or never settles into a single, finite value. If the terms of a series, as we add more and more, just continue without bound or oscillate without stabilizing, we say the series is divergent.
Divergence means the series does not settle down into a finite limit, making it unpredictable and often less useful in calculations.

In the Ratio Test's application phase from the exercise, we would check if the limit of \( \left| \frac{a_{n+1}}{a_n} \right| \) is greater than 1, which would indicate divergence.
However, as we determined that the limit equals 1 for our series, it is neither demonstrated to diverge or converge definitively from this test alone.

Infinite Series Analysis

The study of infinite series involves deciding whether the series converges, diverges, or if further analysis is needed. This process is crucial for working in various fields of mathematics. Applying tests like the Ratio Test allows mathematicians and students to sift through the types of infinite series and understand their behavior.
While the Ratio Test is a powerful tool for series analysis, it’s not always conclusive, as illustrated in our exercise where the limit turned to 1.

This inconclusiveness prompts us to use other methods or tests, like the Root Test or Integral Test, each having unique applicability.
Understanding how to apply and interpret different tests enriches one's capability in analyzing and understanding complex series, thus broadening mathematical problem-solving skills.

• Ratio Test helps compare ratios of sequential terms.
• Inconclusive Ratio Test results point to exploring other tests.
• Analysing series improves comprehension of sums and infinite sequences.