Problem 99 - Essential Calculus

Question

Use the Ratio Test or the Root Test to determine the convergence or divergence of the series. $$ \frac{1}{(\ln 3)^{3}}+\frac{1}{(\ln 4)^{4}}+\frac{1}{(\ln 5)^{5}}+\frac{1}{(\ln 6)^{6}}+\cdots $$

Step-by-Step Solution

Answer

Based on the Ratio Test, the given series is absolutely convergent.

Step 1: Identify the nth term of the series

First, identify the nth term of the series. It is given by: \(a_n = \frac{1}{(\ln n)^{n}}\)

Step 2: Calculate the ratio of (n+1)th term to nth term

Now calculate the ratio of the \((n+1)\)th term to the nth term. The ratio is given by: \(r_n = \frac{a_{n+1}}{a_n} = \frac{(\ln n)^n}{(\ln (n+1))^{n+1}}\)

Step 3: Calculate the Limit

Now, calculate the limit of \(r_n\) as \(n\) approaches infinity. As \(n\) becomes very large, \(\ln (n+1)\) will exceed \(\ln n\), so the ratio \(r_n\) will go to zero. Thus,\(\lim_{{n}\to\infty} {r_n} = 0\)

Step 4: Apply the Ratio Test

The Ratio Test states that if the limit obtained in step 3 is less than 1 (which it is in this case, 0<1), then the series is absolutely convergent.

Key Concepts

Ratio Test

The Ratio Test is a popular method for determining the convergence of a series. It is especially useful when you are working with series of terms that include factorials or powers.
To apply the Ratio Test, follow these steps:

• Identify the general term of the series, denoted by \( a_n \).
• Calculate the ratio \( r_n = \frac{a_{n+1}}{a_n} \).
• Find the limit \( \lim_{{n} \to \infty} r_n \).

The result of this limit will tell you:

• If the limit \( L < 1 \), the series converges absolutely.
• If the limit \( L > 1 \), the series diverges.
• If the limit \( L = 1 \), the test is inconclusive, and you may need to use another test.

In the context of our original exercise, the Ratio Test easily demonstrates the series's convergence because the limit \( \lim_{{n}\to\infty} r_n \) was resolved to be zero, indicating absolute convergence since 0 is less than 1.

Root Test

The Root Test, like the Ratio Test, helps determine the convergence of an infinite series. This test works particularly well with terms that have exponents.
To use the Root Test:

• Start by identifying the general term as \(a_n\) of the series.
• Next, calculate the nth root of the absolute value of \(a_n\) as \(\sqrt[n]{|a_n|}\).
• Determine the limit \(\lim_{{n}\to\infty}\sqrt[n]{|a_n|}\).

The outcome of the Root Test tells us:

• If \(L < 1\), the series converges absolutely.
• If \(L > 1\), it diverges.
• If \(L = 1\), the test is inconclusive, similar to the Ratio Test.

Although the Root Test was not applied in the solution provided, it could be considered as an alternative, especially if the series involves exponential terms or roots that simplify the computation of the limit.

Series Convergence

Series Convergence revolves around determining whether the sum of all terms in a series is finite. A series is a sum of the terms of a sequence.
To evaluate a series, we need a method that reliably checks if the series converges to a certain value or goes off to infinity.
Several tests exist to determine the convergence:

• Ratio Test: Useful for factorials and powers.
• Root Test: Effective for series where terms are raised to powers.
• Comparison Test: Requires comparing a series to a known benchmark series.
• Integral Test: Uses integrals to determine convergence, especially for non-negative terms.

Applying these tests allows us to predict the convergence or divergence of a series, thus helping in evaluating whether a mathematical model or function behaves as expected over an infinite endpoint, as shown by our exercise solution.