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The Integral Test is a helpful tool for determining the convergence of a series. It's especially useful for series where the terms diminish in size in a specific manner, such as those involving logarithmic functions.To apply the Integral Test, you need to ensure the function you're dealing with, say \( f(n) \), meets these criteria:

• It's continuous, meaning no sudden jumps or breaks.
• It's positive for the interval you're considering, which makes things more straightforward.
• It's decreasing, which tells us the terms are becoming progressively smaller.

For the series \( \sum_{n=2}^{\infty} \frac{1}{(\ln n)^{2}} \), we identified \( f(n) = \frac{1}{(\ln n)^{2}} \) as fitting these conditions. This means we can replace the series with an integral: \( \int_{2}^{\infty} \frac{1}{(\ln x)^{2}} \, dx \).If the improper integral converges, the corresponding series also converges. This connection makes the Integral Test a powerful method to analyze series with complex terms.

The natural logarithm function, written as \( \ln(x) \), is a crucial part of mathematical analysis. It's the inverse of the exponential function \( e^x \), where \( e \) is approximately 2.71828. The relation \( y = \ln(x) \) means that \( e^y = x \).Key properties of the natural logarithm function are:

• \( \ln(x) \) is defined for all positive \( x \).
• It's an increasing function, meaning \( \ln(x) \) continues to grow as \( x \) gets larger.
• The logarithm of 1 is 0, i.e., \( \ln(1) = 0 \).

The series \( \sum_{n=2}^{\infty} \frac{1}{(\ln n)^{2}} \) depends on the behavior of \( \ln(n) \). As \( n \) increases, \( \ln(n) \) becomes larger, making \( (\ln(n))^2 \) larger as well, leading to smaller terms in the series. Understanding the nature of \( \ln(x) \) helps us realize why the associated integral can converge, which in turn determines the convergence of the series.

A convergent series is one where the sum of its terms approaches a specific finite value as you add more and more terms. It's a central concept in calculus and mathematical analysis.In the case of the series \( \sum_{n=2}^{\infty} \frac{1}{(\ln n)^{2}} \), we determined its convergence using the Integral Test. By substituting the series with the integral \( \int_{2}^{\infty} \frac{1}{(\ln x)^{2}} \, dx \), we check if the integral achieves a finite result.Steps to determine convergent series:

• Ensure that the integral or test applied confirms a finite sum.
• Once established, conclude the original series also reaches a finite sum.

The finite result from the integral showed us that as \( n \) approaches infinity, the terms of \( \frac{1}{(\ln n)^{2}} \) add up to a specific, bounded sum. This behavior is what characterizes a convergent series, leading to the conclusion that our given series converges.