Uploaded August 2026 | Updated September 2026, 3 weeks ago
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!
#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
Notes and playlists:
- Hive: 3speak.tv/watch?v=mes/example-4-series-with-terms-999
- Notes: peakd.com/mathematics/@mes/infinite-sequences-and-series-the-integral-test-and-estimate-of-sums
- Playlist: youtube.com/playlist?list=PLTXCLnbwZspA
- Sequences and Series: youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz
- MES Links: https://mes.fm/links .
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MES Links: https://mes.fm/links
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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!
#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
Notes and playlists:
- Hive: 3speak.tv/watch?v=mes/example-4-series-with-terms-999
- Notes: peakd.com/mathematics/@mes/infinite-sequences-and-series-the-integral-test-and-estimate-of-sums
- Playlist: youtube.com/playlist?list=PLTXCLnbwZspA
- Sequences and Series: youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz
- MES Links: https://mes.fm/links .
------------------------------------------------------
MES Links: https://mes.fm/links
Donate: https://mes.fm/donate
MES Truth: youtube.com/@mestruth
Official Website: https://MES.fm
Hive: peakd.com/@mes
Email me: contact@mes.fm
Percentage Calculator: https://percentagecalculator.mes.fm
Grade Calculator: https://gradecalculator.mes.fm
BMI Calculator: https://bmicalculator.mes.fm
Mortgage Calculator: https://mortgagecalculator.mes.fm
Timer: https://timer.mes.fm/
