Example 4: Series with terms (ln n)/n diverges by the Integral Test

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In this video, I show that the sum of the series with terms (ln n) / n diverges to infinity by computing its corresponding improper integral via the Integral Test. First noting that the natural logarithm function is continuous, and positive for x is greater than 1, then so is the corresponding function f(x) = (ln x) / x. However, it is uncertain whether the function is increasing or decreasing, so I take the derivative and show that it is negative for x is greater than the number e, thus the function itself is also decreasing. This allows us to apply the Integral Test for convergence and the corresponding improper integral approaches infinity, thus our series diverges!

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#math #calculus #series #logarithms #education

Timestamps:

• Example 4: Series with terms (ln n)/n – 0:00
• Solution: Compute derivative to see if the terms are decreasing – 0:18
• Recap on inverse functions and natural log – 3:43
• ln x is greater than 1 for x is greater than e – 7:20
• Derivative is negative when x is greater than e – 8:00
• Apply the Integral Test since the terms are ultimately decreasing – 9:01
• Improper integral diverges, so the series diverges by the Integral Test – 12:45

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