# A harmonic series without the nines

• I
• Thecla
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.
Thecla
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?

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.

## 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.

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