Fraction sums

[Agent Smith]

I performed the following procedure.

1. Take a fraction, [imath]\frac{a}{b} \text{ such that } 0 < \frac{a}{b} < 1[/imath]
2. [imath]\frac{a}{b} + \frac{9a}{10b} + \frac{8a}{10b} + \dots + \frac{2a}{10b} + \frac{1a}{10b}[/imath]
3. At each addition step above, I checked if the partial sum was approaching a limit. This I did by comparing each fraction above to a preset value like [imath]0.001[/imath] or [imath]0.0000001[/imath]
4. If the sum of the fractions [imath]\geq 1[/imath], I stopped.
N.B if neither of the conditions (the sum [imath]\geq 1[/imath] or the fraction so taken is above the preset value [imath]0.001[/imath] or [imath]0.0000001[/imath]) are fulfilled we repeat the procedure on the smallest fraction in the list viz. [imath]\frac{1a}{10b}[/imath] i.e. [imath]\frac{9a}{100b} + \frac{8a}{100b} + \dots + \frac{2a}{100b} + \frac{1a}{100b}[/imath]

I found out that the procedure performed on the fraction [imath]0.05[/imath] yields a limit viz. [imath]0.2999(7)[/imath]

For some fractions the sum [imath]> 1[/imath].

I'm doing this by trial and error and I think I'm looking for a fraction [imath]x[/imath] such that the [imath]\text{sum} = 1[/imath]. Does [imath]x[/imath] exist and if it does what is [imath]x[/imath]?

Gracias.

 

[mario99]

I performed the following procedure.

1. Take a fraction, [imath]\frac{a}{b} \text{ such that } 0 < \frac{a}{b} < 1[/imath]
2. [imath]\frac{a}{b} + \frac{9a}{10b} + \frac{8a}{10b} + \dots + \frac{2a}{10b} + \frac{1a}{10b}[/imath]
3. At each addition step above, I checked if the partial sum was approaching a limit. This I did by comparing each fraction above to a preset value like [imath]0.001[/imath] or [imath]0.0000001[/imath]
4. If the sum of the fractions [imath]\geq 1[/imath], I stopped.
N.B if neither of the conditions (the sum [imath]\geq 1[/imath] or the fraction so taken is above the preset value [imath]0.001[/imath] or [imath]0.0000001[/imath]) are fulfilled we repeat the procedure on the smallest fraction in the list viz. [imath]\frac{1a}{10b}[/imath] i.e. [imath]\frac{9a}{100b} + \frac{8a}{100b} + \dots + \frac{2a}{100b} + \frac{1a}{100b}[/imath]

I found out that the procedure performed on the fraction [imath]0.05[/imath] yields a limit viz. [imath]0.2999(7)[/imath]

For some fractions the sum [imath]> 1[/imath].

I'm doing this by trial and error and I think I'm looking for a fraction [imath]x[/imath] such that the [imath]\text{sum} = 1[/imath]. Does [imath]x[/imath] exist and if it does what is [imath]x[/imath]?

Gracias.

[imath]\displaystyle\frac{2}{11}\sum_{k=1}^{10}\frac{11-k}{10} = 1[/imath]

[imath]\displaystyle x = \frac{a}{b} = \frac{2}{11}[/imath]

 

[Agent Smith]

[imath]\frac{1}{2} + \left[\frac{1}{4} + \frac{1}{8} + \dots \right] = 1[/imath]

 

[mario99]

[imath]\frac{1}{2} + \left[\frac{1}{4} + \frac{1}{8} + \dots \right] = 1[/imath]

What is your question?

I am sure you know that the geometric series always converges for [imath]|r| < 1[/imath].

See what happens when [imath]|r| > 1[/imath]:

[imath]2 + 4 + 8 + \dots = \infty[/imath]

 

[Agent Smith]

What is your question?

I am sure you know that the geometric series always converges for [imath]|r| < 1[/imath].

See what happens when [imath]|r| > 1[/imath]:

[imath]2 + 4 + 8 + \dots = \infty[/imath]

I'm looking for convergence towards unity/1. This [imath]\frac{1}{2} + \left[\frac{1}{4} + \frac{1}{8} + \dots \right][/imath] is one. I know a little bit of geometric means and I kinda sorta get your point.

 

[mario99]

I'm looking for convergence towards unity/1. This [imath]\frac{1}{2} + \left[\frac{1}{4} + \frac{1}{8} + \dots \right][/imath] is one. I know a little bit of geometric means and I kinda sorta get your point.

What do you mean by that?

You have already had it:

[imath]\displaystyle \sum_{k=1}^{\infty} \frac{1}{2^k} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots = 1[/imath]