Problems Plus 22: Spiraling Triangle Series

In this video I construct an infinite series of right triangles rotated about a single point and show that the central angle diverges to infinity as the number of triangles approach infinity. To show this, I first construct a table determining the base lengths of each triangle using the Pythagorean Theorem to obtain a formula for the n-th triangle base. From this we can obtain the angle as a formula involving the inverse tangent trigonometric function. Applying the Limit Comparison Test with a diverge p-series, I show, using L'Hospital's Rule, that the angle series diverges, hence the angle approaches infinity as the number of triangles approach infinity. This means that the series of triangles makes infinitely many turns around the central point.

The timestamps of key parts of the video are listed below:

- Problem 22: Spiraling Triangles: 0:00
- Solution: Repeating Pythagorean Theorem: 1:16
- Formula for n-th Triangle: 5:16
- Limit Comparison Test: 8:57
- Comparing with a Divergent p-series: 10:21
- Applying the Limit Comparison Test with our original series: 14:36
- Recap on tan(x): 18:59
- Applying L'Hospital's Rule: 20:56
- The angle of the triangles diverges to infinity: 22:30
- Recap on the derivative of arctan(x): 23:11

This video was taken from my earlier video listed below:

- Infinite Sequences and Series: Problems Plus: https://youtu.be/zjdkQIIdTbg
- HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-problems-plus
- Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FQ96Egr5R7fZGeDTIUKz8P

Related Videos:

Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz .

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