Uploaded June 2026 | Updated September 2026, 3 weeks ago
In this video, I go over the geometric series, which is the sum a + a r + a r^2 + ..., and show that if the absolute value of the common ratio r is less than 1, then the sum is just the first term divided by one minus the common ratio. I first derive this formula by a very simple yet genius method of multiplying the infinite series by r, and then subtracting the resulting series and rearranging to solve for the sum. I also illustrate this geometrically via similar triangles. For all other values of r, the series is divergent, hence does not exist.
#math #calculus #series #geometricseries #education
Timestamps:
- Example 1: Geometric series multiplies previous term by common ratio r – 0:00
- If r = 1, then the series diverges – 2:07
- If r ≠ 1, then we can obtain a formula for the sum of the series using an ingenious method of multiplying by r – 2:58
- If r is between -1 and 1, then the series r^n converges to zero – 5:53
- Sum of geometric series for this case is just the first term divided by (1 - r) – 7:30
- If |r| is greater than 1, then the geometric series is divergent – 8:24
- Summary of Geometric Series – 9:09
- Geometric demonstration of the geometric series using similar triangles – 10:33
- Obtain the same formula: Sum = first term divided by common ratio – 14:05
Notes and playlists:
- 3Speak: 3speak.tv/watch?v=mes/in-this-video-i-go-over-the-544
- Hive Notes: peakd.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more
- Playlist: youtube.com/playlist?list=PLai3U8-WIK0FfxN_I9trdgSnWtJS-6wK1
- Sequences and Series: youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz
- MES Links: https://mes.fm/links .
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In this video, I go over the geometric series, which is the sum a + a r + a r^2 + ..., and show that if the absolute value of the common ratio r is less than 1, then the sum is just the first term divided by one minus the common ratio. I first derive this formula by a very simple yet genius method of multiplying the infinite series by r, and then subtracting the resulting series and rearranging to solve for the sum. I also illustrate this geometrically via similar triangles. For all other values of r, the series is divergent, hence does not exist.
#math #calculus #series #geometricseries #education
Timestamps:
- Example 1: Geometric series multiplies previous term by common ratio r – 0:00
- If r = 1, then the series diverges – 2:07
- If r ≠ 1, then we can obtain a formula for the sum of the series using an ingenious method of multiplying by r – 2:58
- If r is between -1 and 1, then the series r^n converges to zero – 5:53
- Sum of geometric series for this case is just the first term divided by (1 - r) – 7:30
- If |r| is greater than 1, then the geometric series is divergent – 8:24
- Summary of Geometric Series – 9:09
- Geometric demonstration of the geometric series using similar triangles – 10:33
- Obtain the same formula: Sum = first term divided by common ratio – 14:05
Notes and playlists:
- 3Speak: 3speak.tv/watch?v=mes/in-this-video-i-go-over-the-544
- Hive Notes: peakd.com/mathematics/@mes/infinite-series-definition-examples-geometric-series-harmonics-series-telescoping-sum-more
- Playlist: youtube.com/playlist?list=PLai3U8-WIK0FfxN_I9trdgSnWtJS-6wK1
- Sequences and Series: youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz
- MES Links: https://mes.fm/links .
------------------------------------------------------
Become a MES Super Fan! 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: 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
