The Integral Test + Example on Convergence of a Series @mes

The Integral Test + Example on Convergence of a Series @mes

Uploaded August 2026 | Updated September 2026, 3 weeks ago

In this video, I go over the Integral Test, which states that a series converges only if its corresponding improper integral also converges. Likewise, if the improper integral diverges, then the series also diverges. This stems from my earlier introduction video illustrating how the sum of an infinite series is directly tied to the area under the curve of the equation for the terms of the series. I illustrate this with an example on the series with terms 1/(n²+1), which converges since its improper integral converges and equals π/4.

#math #calculus #series #integral #trigonometry

Timestamps:

- The Integral Test – 0:00
- It is not necessary to start the integral or series at n = 1 – 2:05
- It is not necessary for the function to be always decreasing, only that it is ultimately decreasing – 2:54
- Example 1: Test convergence of series with terms 1/(n²+1) – 5:09
- Recall derivative of inverse tangent – 6:28
- Recall exact trigonometric ratios for tangent – 8:29
- Recall the graph for tangent and inverse tangent – 10:08
- Improper integral is convergent, thus the series is convergent by the Integral Test – 14:12

Notes and playlists:

- Hive: 3speak.tv/watch?v=mes/the-integral-test--example-744
- 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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The Integral Test + Example on Convergence of a Series

MES Livestream 115: John Hutchison Joins the Show Again!

$Telescoping Sum: All the terms of this series cancel out except for the first and last term In this video, I go over an infinite series in which arises an example of the famous telescoping sum such that all the terms of the series cancel except for the first and last term. The series with terms 1/(n(n+1)) is not a geometric series, so we have to start off with the definition of a convergent series and begin by writing out the terms of its n-th partial sum. We can then simplify the terms of the partial sum by using partial fraction decomposition. This yields the terms 1/n - 1/(n+1), which results in all of the middle terms to cancel out in pairs, leaving just 1 - 1/(n+1). Taking the limit as n approaches infinity, we obtain our sum is equal to 1. The telescoping sum gets its name from the collapsing telescope (the ones pirates had). #math #calculus #series #TelescopingSum #education Timestamps: - Example 6: Find the sum if this series is convergent – 0:00 - Solution: This is not a geometric series, so go back to the definition of a convergent series and compute the partial sums – 0:17 - Simplify the partial sum via partial fraction decomposition – 1:55 - Notice that all the terms in the partial sum cancel except for the first and last! – 5:05 - This is named after a collapsing telescope (like how pirates had) – 7:49 - The limit of the partial sums is equal to 1 – 8:22 - Graph showing the series terms approach zero as the partial sums approach 1 – 9:44 Notes and playlists: - 3Speak: https://3speak.tv/watch?v=mes/telescoping-sum-all-the-ter-829 - Hive Notes: https://peakd.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more - Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FfxN_I9trdgSnWtJS-6wK1 - Sequences and Series: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz - MES Links: https://mes.fm/links . Become a MES Super Fan! https://www.youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate SUBSCRIBE via EMAIL: https://mes.fm/subscribe MES Links: https://mes.fm/links MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps$ $Vortex Math: Summing the Digits of a Number is just the Remainder In this video I show that the vortex sum of a number (summing up the digits until obtaining a single digit is obtained) is just the remainder when dividing the number by 9. I go over a couple of examples while also illustrating why this is the case by writing the transition numbers 10, 100 as (9 + 1), (99 + 1). I will prove this is always the case when in later videos when I cover the modulo function. Note that I denote v= when taking the vortex sum. #math #vortexmath #modulararithmetic #algebra #education Timestamps: - Summing integer digits and the remainder after division. v= = vortex sum – 0:00 - 11 = 10 + 1 = (9 + 1) + 1 = 9 + (1 + 1) = 9 + 2 – 0:55 - Dividing 11 by 9 we get 1.222... – 1:32 - The 2 is the remainder – 2:55 - 678 v= 6 + 7 + 8 = 21 v= 2 + 1 = 3. 678/9 = 75.333... – 3:25 - Deriving in MES fraction form, separating transition 10 numbers into 9 + 1 – 4:10 - Factoring out the 9s in the denominator – 6:10 - Repeat division of 21/9 – 8:15 - Remainder is always the same as the vortex sum – 10:41 Notes and playlists: - Summary: https://inleo.io/threads/view/mes/re-leothreads-2stpi3pdw - Notes: https://peakd.com/hive-128780/@mes/messcience-2-vortex-math-part-1-number-theory-and-modular-arithmetic - Vortex Math playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EbRnMsUBx2RxlerL7GQuLX - MES Science playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0GhjCHmTw1XbqMD_EdVKdd9 . Become a MES Super Fan! https://www.youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate SUBSCRIBE via EMAIL: https://mes.fm/subscribe MES Links: https://mes.fm/links MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps$

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$Definition and Notation of Infinite Series In this video, I go over infinite series (or just series), which is defined as the summation of the terms of an infinite sequence of numbers. If we add an infinite sequence of numbers whose terms are each greater than 1, then the sum goes to infinity. But if the terms are fractions, then in some cases the sum tends towards a finite limit. In this case, we obtain a convergent series — otherwise, if there is no finite sum, then the series is called divergent. A partial sum is the sum of a specific number of terms of a series. The actual sum is the limit of the partial sums as the terms approach infinity. This is similar to how the improper integral is defined. #math #calculus #series #sequences #education Timestamps: - Adding terms of an infinite sequence is called an infinite series or just a series – 0:00 - Adding infinite fractions of factors of 2 – 1:52 - Excel table calculating partial sums of the series and showing it approaches 1: https://1drv.ms/x/s!As32ynv0LoaIh8gwtUIvFFz4HKE0DA – 3:33 - If the partial sum of a series has a limit, then the series has a finite sum – 6:09 - Definition 1: A series is convergent if it has a sum; otherwise, it is divergent – 6:53 - Comparing infinite series to improper integrals – 9:47 Notes and playlists: - 3Speak: https://3speak.tv/watch?v=mes/2957acdd - Summary: https://inleo.io/threads/view/mes/re-leothreads-cytxwfsu - Hive Notes: https://peakd.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more - Sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FfxN_I9trdgSnWtJS-6wK1 - Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz . Become a MES Super Fan! https://www.youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate SUBSCRIBE via EMAIL: https://mes.fm/subscribe MES Links: https://mes.fm/links MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps$

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The Integral Test + Example on Convergence of a Series