Name: Problems Plus 9: Taking Multiple Derivatives of the Geometric Series
Uploaded: 2023-09-19T03:03:08+00:00
Duration: 21 min 22 s
Description: In this video I go over the sum of an infinite series that involves taking multiple derivatives of the Geometric Series. Starting off with the Geometric Series for x^n, we can apply the derivative to

2 years ago

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math calculus science geometric series quotient rule power series derivatives sequences test for convergence test for divergence

In this video I go over the sum of an infinite series that involves taking multiple derivatives of the Geometric Series. Starting off with the Geometric Series for x^n, we can apply the derivative to it for its radius of convergence, as derived from my earlier video on Power Series. When taking derivatives, the Radius of Convergence remains the same but we still have to determine the endpoints separately. The series with the terms (n^3)x^n has a radius of convergence equal to 1 and diverges at the endpoints, that is it diverges at x = 1 and x = -1.

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

- Problem 9: 0:00
- Solution: Start with a Geometric Series: 0:20
- Differentiating with Quotient Rule: 6:42
- Radius of Convergence is 1: 16:40
- Checking for convergence at the endpoints: 18:18
- Endpoints diverge: 21:09

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