# Exploring the Relationship Between a/b and Geometric Series

• I
• Jehannum
In summary, a probability problem led to the discovery of the relationship $$\frac a b = \frac a {(b-1)} - \frac a {{(b-1)}^2} + \frac a {{(b-1)}^3} - \frac a {{(b-1)}^4} + ~...$$ which can be simplified to a geometric series. The Cesaro sum of this series is 1/2, but it only converges for values of b outside the range of 0 to 2. Further research on similar series may lead to interesting results.
Jehannum
While working on a probability problem I accidentally found this relationship:

$$\frac a b = \frac a {(b-1)} - \frac a {{(b-1)}^2} + \frac a {{(b-1)}^3} - \frac a {{(b-1)}^4} + ~...$$
I have done a bit of work on it myself, and have tried to research similar series. It seems to lead to some interesting results. For example, when a = 1 and b = 2 it doesn't work because you get 1 - 1 + 1 - 1 + 1 ... but it's interesting that the Cesaro sum of this series is 1/2.

Can anyone provide links or information on anything relevant?

Jehannum said:
While working on a probability problem I accidentally found this relationship:

$$\frac a b = \frac a {(b-1)} - \frac a {{(b-1)}^2} + \frac a {{(b-1)}^3} - \frac a {{(b-1)}^4} + ~...$$
I have done a bit of work on it myself, and have tried to research similar series. It seems to lead to some interesting results. For example, when a = 1 and b = 2 it doesn't work because you get 1 - 1 + 1 - 1 + 1 ... but it's interesting that the Cesaro sum of this series is 1/2.

Can anyone provide links or information on anything relevant?
Taking out the common factor of ##a## and letting ##x = \frac{1}{b - 1}##, you have a geometric series:
$$S = x - x^2 + x^3 - x^4 + \dots$$This converges for ##|x| < 1## to ##S = \frac{x}{1+ x}##, and a bit of algebra shows that indeed:$$\frac{x}{1+ x} = \frac 1 b$$And ##|x| < 1## implies ##b < 0## or ##b > 2##. In particular, this series does not converge for ##b = 2##.

nuuskur, mfb and berkeman

## 1. Is this series based on real scientific concepts?

It depends on the specific series. Some are based on real scientific concepts and theories, while others may take creative liberties for entertainment purposes.

## 2. Are the scientific elements accurate in this series?

Again, it varies. Some series strive for accuracy and consult with experts, while others may prioritize entertainment over accuracy.

## 3. Can I learn anything new about science from watching this series?

Possibly. Many series incorporate educational elements and aim to teach viewers about scientific concepts in an engaging way.

## 4. Is this series appropriate for all ages?

It depends on the series and its intended audience. Some may be geared towards children, while others may contain mature content and themes.

## 5. Will I enjoy this series if I'm not a science enthusiast?

It's difficult to say for sure, as enjoyment is subjective. However, many series aim to appeal to a wide audience and may incorporate elements beyond just science.

• General Math
Replies
7
Views
1K
• General Math
Replies
5
Views
2K
• General Math
Replies
3
Views
960
• General Math
Replies
3
Views
2K
• General Math
Replies
17
Views
4K
• General Math
Replies
6
Views
1K
• General Math
Replies
1
Views
2K
• Calculus
Replies
3
Views
6K
• Introductory Physics Homework Help
Replies
28
Views
602
• General Math
Replies
1
Views
885