Question

In Exercises 97-100, use the Ratio Test or the Root Test to determine the convergence or divergence of the series. $$ 1+\frac{1 \cdot 2}{1 \cdot 3}+\frac{1 \cdot 2 \cdot 3}{1 \cdot 3 \cdot 5}+\frac{1 \cdot 2 \cdot 3 \cdot 4}{1 \cdot 3 \cdot 5 \cdot 7}+\cdots $$

Short answer

The given series is convergent.

Step-by-step solution

Write the General Term of the Series

Look at the pattern of the given series and write the general term. The series can be expressed on the form: \[ a_{n}=\frac{1\cdot 2\cdot 3\cdot...\cdot n}{1\cdot3\cdot5\cdot...\cdot(2n-1)}\]

Apply the Ratio Test

The Ratio Test can be used to determine if a series converges or diverges by looking at the limit as \(n\) approaches infinity for the absolute value of the ratio of the \((n+1)\)-th term to the \(n\)-th term. So we need to calculate \[\lim_{n\to\infty} \left |\frac{a_{n+1}}{a_{n}}\right |\], where \(a_{n+1}\) and \(a_{n}\) are the sequential terms of the series. So \[\lim_{n\to\infty} \left|\frac{1\cdot2\cdot3\cdot...\cdot(n+1)}{1\cdot3\cdot...\cdot(2n+1)} \div \frac{1\cdot2\cdot3\cdot...\cdot n}{1\cdot3\cdot...\cdot(2n-1)}\right|\]

Calculate the Limit

Simplify the limit expression by canceling out similar terms. It will yield: \[\lim_{n\to\infty} \left|\frac{(n+1)}{(2n+1)}\right|\] Now, take the limit as \(n\) approaches infinity in it. This will yield the result to be \[\lim_{n\to\infty} \frac{(n+1)}{(2n+1)} = \frac{1}{2}\]

Determine the Convergence/Divergence of the Series

In the Ratio Test, a series \(\Sigma a_n\) will converge if the limit as \(n\) approaches infinity for the absolute value of the ratio of any two successive terms is less than 1. Here, our limit as \(n\) approaches infinity for the absolute value of the ratio of the \((n+1)\)-th term to the \(n\)-th term is \(\frac{1}{2}\), which is less than 1. Hence, the given series is convergent.

Key concepts

Ratio Test

When trying to understand whether an infinite series converges or diverges, the Ratio Test is a powerful tool in a student's mathematical arsenal.
To apply the Ratio Test, we look at the sequence of absolute ratios of consecutive terms, namely \( \left| \frac{a_{n+1}}{a_n} \right| \), as \( n \) approaches infinity. If this limit is less than 1, the series converges; if it's more than 1, the series diverges; and if it equals 1, the test is inconclusive.
For the series in our exercise, the Ratio Test simplified to \( \lim_{n\to\infty} \left| \frac{(n+1)}{(2n+1)} \right| = \frac{1}{2} \), which clearly indicates convergence since \( \frac{1}{2} < 1 \).

• Identify the general term \( a_n \) of the series.
• Calculate the limit of \( \left| \frac{a_{n+1}}{a_n} \right| \) as \( n \) approaches infinity.
• Apply the Ratio Test rule for convergence (\( < 1 \) implies convergence).

By understanding these steps, learners can confidently use the Ratio Test to analyze series.

Root Test

Another method to determine the convergence of an infinite series is the Root Test, especially useful in certain situations where the Ratio Test is inconclusive or difficult to apply.
This test involves taking the \( n \)th root (where \( n \) tends towards infinity) of the absolute value of the \( n \)th term, \( \lim_{n\to\infty} \sqrt[n]{|a_n|} \). Similar to the Ratio Test, if this limit is less than 1, the series converges; if it is greater than 1, the series diverges; and if it equals 1, the test is inconclusive.
The Root Test can be particularly useful when each term of the series is raised to a power, making the root extraction process more natural.

• Compute the \( n \)th root of the absolute value of the \( n \)th term \( (\sqrt[n]{|a_n|}) \).
• Find the limit of this expression as \( n \) approaches infinity.
• Use the criterion of the Root Test to conclude about convergence.

With practice, the Root Test becomes another indispensable technique in analyzing series for convergence.

Convergence of Series

The concept of whether a series converges or diverges is key to understanding its behavior. A series is considered to converge if its partial sums approach a finite limit as we add more terms. Otherwise, it diverges.
In the context of our exercise converging means that as you add up terms of the series indefinitely, the sum gets closer and closer to a specific value. Divergence, on the other hand, implies that the sum grows without bound or oscillates without approaching a specific value.
Convergence is not only about whether the series adds up to a number but also about the nature of the function it represents; it can suggest stability and predictability.

Key Indicators of Convergence

• Finiteness of the limit of partial sums.
• Behavior of terms getting progressively smaller.
• Application of specific tests like Ratio or Root Tests yielding a limit less than 1.

A solid grasp of this concept enables learners to dissect complex mathematical questions and analyze the functionality and continuity of series-derived functions.

Limit of a Sequence

The limit of a sequence is a fundamental concept in calculus that pertains to the value a sequence 'approaches' as the index \( n \) becomes very large. If a sequence has a limit, then as \( n \) grows, the terms of the sequence become arbitrarily close to the limit value.
In the provided solution, we are interested in the limit of the sequence of ratios, \( \lim_{n\to\infty} \frac{a_{n+1}}{a_n} \), which informs us about the behavior of the series as a whole. If the limit exists and is finite, it tells us that the terms of the sequence are stabilizing to a certain value, which leads us to make conclusions about series convergence.

• Analyze how the sequence behaves as \( n \) gets larger.
• Determine if the sequence terms are approaching a specific value.
• Use the limit concept to draw conclusions about the series.

The ability to calculate and understand limits is essential for students to progress in calculus, as it forms the basis for defining continuity, convergence, and even fundamental concepts of derivatives and integrals.