In this video I go over more advanced tests for convergence in order to determine the convergence or divergence of the Binomial Series at the endpoints. In particular, I first derive Kummer’s Test, which is a generalization of both the Ratio Test, covered in my earlier video, and Raabe’s Test, which I cover in this video. I also show that Raabe’s Test is essentially a test that compares series to the p-series, which I had covered before.
In my earlier video, I had covered the Binomial Series and showed using the Ratio Test that the radius of convergence was less than 1 but the test was inconclusive at the endpoints. This is where I use Raabe’s Test to overcome the shortcomings of the Ratio Test and show that at the endpoints, the Binomial Series may converge or diverge depending on the value of k. This is a pretty epic and extensive proof video, and likely far more thorough than any other proof found online, so make sure to watch it!
Note that Kummer’s Test was first obtained by German mathematician Ernst Eduard Kummer, while Raabe’s Test was obtained by Swiss mathematician Joseph Ludwig Raabe, and both during the mid-1800s.
Also note, that I obtained the proofs for Kummer’s Test and Raabe’s Test from a paper by Sarah Fix from Whitman College, USA. As for the Binomial Series proof, I used mainly the lecture PDF by John Labute from McGill University, Canada.
The topics covered in this video are listed below with their time stamps.
Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIiL4occLuFG9lN_2stg?e=jdv8Up
View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/kummer-s-test-raabe-s-test-and-convergence-of-binomial-series
Sequences and Series Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz
Infinite Sequences and Series: Power Series: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-power-series
Infinite Sequences and Series: Representations of Functions as Power Series: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-representations-of-functions-as-power-series
Infinite Sequences and Series: Taylor and Maclaurin Series: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-taylor-and-maclaurin-series
Infinite Sequences and Series: The Comparison Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-the-comparison-tests
Infinite Sequences and Series: Alternating Tests: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-alternating-tests
Infinite Sequences and Series: Strategy for Testing Series: https://peakd.com/mathematics/@mes/infinite-sequences-and-series-strategy-for-testing-series .
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