Problems Plus 6: Sum of Reciprocals of Positive Integers Whose Prime Factors are 2s and 3s

8 months ago

In this video I go over the sum of the series whose terms are reciprocals of the positive integers whose only prime factors are 2s and 3s. Recall that a prime number is a number that can only be divided cleanly by itself or 1, for example the numbers 2, 3, and 5. A prime factor is a prime number that can divide into a number cleanly, for example 3 and 2 are prime factors of the number 6. A reciprocal is a number of the form 1/x. When writing out such a series of reciprocals, I show that we can formulate it as a double summation, where one summation is for the 1/2 terms and the other summation is for the 1/3 terms. Separating these further, I show they are indeed convergent geometric series. This means we can simply apply the geometric series summation formula and finally obtain the number 3. A truly fascinating result to what at first seemed a very convoluted problem!

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

Problem 6: 0:00
Solution: 0:36
- Recap on Prime Factors and Prime Numbers: 0:45
- Recap on Reciprocals: 1:45
- Back to the Solution: 1:57
- Double Summation: 4:54
- Two convergent Geometric Series: 7:53
- Sum is equal to 3: 9:01

This video was taken from my earlier video listed below: