Apply the integral test to determine if the series converges or diverges and make sure it satisfies the three conditions.
Transcribed Image Text:The image displays the mathematical expression of an infinite series:
\[
\sum_{n=2}^{\infty} \frac{1}{n \cdot (\ln n)}
\]
### Explanation:
- The symbol \(\sum\) denotes the summation, representing the sum of a sequence of terms.
- The expression runs from \(n = 2\) to \(\infty\), indicating that the sequence starts at \(n = 2\) and continues indefinitely.
- Each term in the sequence is in the form of \(\frac{1}{n \cdot (\ln n)}\):
- \(n\) represents the natural numbers starting from 2.
- \(\ln n\) is the natural logarithm of \(n\).
- The denominator is the product of \(n\) and its natural logarithm, thus expressing this specific form of harmonic-like series.
This series can be analyzed for convergence properties using techniques from calculus.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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![The image displays the mathematical expression of an infinite series:
\[
\sum_{n=2}^{\infty} \frac{1}{n \cdot (\ln n)}
\]
### Explanation:
- The symbol \(\sum\) denotes the summation, representing the sum of a sequence of terms.
- The expression runs from \(n = 2\) to \(\infty\), indicating that the sequence starts at \(n = 2\) and continues indefinitely.
- Each term in the sequence is in the form of \(\frac{1}{n \cdot (\ln n)}\):
- \(n\) represents the natural numbers starting from 2.
- \(\ln n\) is the natural logarithm of \(n\).
- The denominator is the product of \(n\) and its natural logarithm, thus expressing this specific form of harmonic-like series.
This series can be analyzed for convergence properties using techniques from calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb4f8469-eda0-4bfd-b67e-0d596825d7c6%2F3a3ba4e1-4feb-45ec-8515-736ab9f946d7%2Fqj69zq_processed.png&w=3840&q=75)
