A harmonic series without the nines

In summary, the harmonic series with terms that contain the number nine is infinite, but excluding those terms results in a sum just under 23. The percentage of excluded terms increases as the range of numbers increases, with 19% excluded in the range of 1 to 100 and 27.1% excluded in the range of 1 to 1000. When considering N digit numbers, there are 10N numbers with up to N digits and 9N numbers with up to N digits but no 9, leading to a simple equation for the fraction of numbers with and without 9.
  • #1
Thecla
132
10
TL;DR Summary
The sum of an harmonic series without numbers containing a nine is finite
The sum of the harmonic series(1/1+1/2+1/3...) is infinite. However, if you exclude all the terms that contain the number nine, the sum is just under 23.
From 1 to 100 19% of the terms are excluded
From 1 to 1000 27.1% of the terms are excluded
Is there a formula for a N digit number what the percentage of numbers from 1 to N that contain a 9?
 
Mathematics news on Phys.org
  • #2
There are 10N numbers with up to N digits (or exactly N digits if we use leading zeros) because every digit has 10 options. There 9N numbers with up to N digits but no 9 for the same reason. That leads to a simple equation for the fraction of numbers with/without 9 up to N digits.
 

Related to A harmonic series without the nines

1. What is a harmonic series without the nines?

A harmonic series without the nines is a mathematical series in which the denominators of each term are consecutive integers, but the number nine is excluded from the series. For example, a harmonic series without the nines would be: 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/10, 1/11, 1/12, etc.

2. Why are the nines excluded from this series?

The exclusion of the nines from this series is a mathematical curiosity that has no practical application. It is often used as a puzzle or exercise in math classes to challenge students to think critically about patterns and sequences.

3. What is the sum of a harmonic series without the nines?

The sum of a harmonic series without the nines is infinite. This means that the series continues on forever without ever reaching a finite value. This is due to the fact that as the denominators increase, the terms in the series get closer and closer to zero, but never actually reach zero.

4. How is a harmonic series without the nines different from a regular harmonic series?

A regular harmonic series includes all integers in the denominators, while a harmonic series without the nines excludes the number nine. This small difference changes the pattern of the series and makes it more challenging to calculate the sum.

5. What is the significance of a harmonic series without the nines?

A harmonic series without the nines has no real significance in mathematics or in practical applications. It is simply a mathematical curiosity that can be used as a tool to teach students about patterns and sequences. However, it does have some interesting properties that have been studied by mathematicians.

Similar threads

Replies
8
Views
972
Replies
4
Views
534
  • General Math
Replies
7
Views
1K
Replies
3
Views
3K
Replies
1
Views
2K
Replies
20
Views
1K
Replies
3
Views
996
Replies
68
Views
9K
  • General Math
Replies
2
Views
1K
Replies
9
Views
1K
Back
Top