Problems Plus 26: Partial Sum of Harmonic Series is NOT an Integer

In this video I go over a very unique problem on showing that the n-th partial sum of the harmonic series is NOT an integer. The harmonic series is the sum of the terms 1/i where i is a positive integer. To show that the partial sum is not an integer, we require some truly outside-of-the-box thinking, which fortunately for us is made easier with the given hint. To prove this, we first assume that the partial sum IS an integer and then observe what happens to 2 similar equations. The first is the multiplication of (the product of all odd integers less than or equal to n) * (the largest power of 2 that is less than or equal to n) * (partial sum). The second equation is the same as the first but with the assumption that the partial sum is an integer. After going over some very unique mathematical reasoning, I show that the left side of the equation is always odd while the right side is always even. This is a contradiction and thus the partial sum can not be an integer! Pretty epic brain twister!

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

- Problem 26: Partial Sum of Harmonic Series is NOT an Integer: 0:00
- Solution: 2:27
- Right side of hint equation is even: 4:16
- Left side of hint equation is odd: 7:23
- Checking when r is less than k: 13:41
- Checking when r = k: 16:56
- Summary: Partial Sum of Harmonic Series is NOT an integer: 20:45

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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