Problems Plus 23: Sum of Series of Reciprocals of Positive Integers Without the Digit 0

In this video I show that the series whose terms are reciprocals of positive integers, excluding the digit 0, has a sum that is less than 90. This is a very interesting problem as it shows how seemingly arbitrary series can have very simple solutions. To solve this problem, I first group the terms of the series such that each group has the same number of digits in the denominator. Next, I count the number of terms in each grouping and find a general formula for the n-th group. I also note that each term (besides the first one) is less than 1/10^n-1. From this we can write a formula for the sum of the infinite series, which turns out to be a convergent geometric series. This means we can simply use our formula for the convergent geometric series and obtain our final answer, which is the series is less than 90. Epic stuff!

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

• Problem 23: Sum of reciprocals of positive integers without the digit 0: 0:00
• Solution: Group the terms into number of digits: 0:25
• Each term is less than 1/10^n-1: 5:26
• Max sum of each grouping: 7:18
• Sum of series is less than a convergent geometric series: 9:06
• Sum of series is less than 90: 10:35

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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