- #1
lukepeterpaul
- 14
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hi, i would be very grateful if someone could start me off me in the right direction in numbering the rationals such that the infinite sum of (x_n - x_n+1)^2 (where x_n are rationals) converges.
thanks!
thanks!
I think you mean [tex]\sum (x_n-x_{n+1})^2[/tex].lukepeterpaul said:[tex]\sum(x_n - x_n+1)^2[/tex] over n, where n ranges from 1 to infinity
this is all the info I've got...
lukepeterpaul said:hi, i would be very grateful if someone could start me off me in the right direction in numbering the rationals such that the infinite sum of (x_n - x_n+1)^2 (where x_n are rationals) converges.
thanks!
dimitri151 said:The sum can not converge as I've understood the problem. If you include the integers then there will be an infinite number of terms which are greater than or equal to one and the series can't converge. Do you mean the rationals in (0,1) or something like that?
lukepeterpaul said:jarle, thanks for your solution. however, would i be right in thinking that the proposed way of enumerating the rationals does not cover all the rationals? e.g. y1+1/1000, say, won't be included...? i apologize if I'm mistaken...
The purpose of "Numbering the Rationals for Convergence" is to provide a systematic way to order and manipulate rational numbers in order to calculate the convergence of infinite sums. This can be particularly useful in fields such as mathematics, physics, and engineering.
Traditional methods of calculating infinite sums often involve approximations and estimations, which can lead to errors and imprecise results. "Numbering the Rationals for Convergence" offers a more precise and systematic approach by assigning a unique index to each rational number, allowing for more accurate calculations and comparisons.
Yes, "Numbering the Rationals for Convergence" can be applied to all types of infinite sums, including both convergent and divergent series. It provides a general framework for organizing and evaluating rational numbers, making it applicable to a wide range of infinite sums.
No, prior knowledge of advanced mathematics is not required to understand "Numbering the Rationals for Convergence". While some familiarity with rational numbers and basic algebra may be helpful, the guide is designed to be accessible to a wide range of readers, including students and professionals in various fields.
Yes, "Numbering the Rationals for Convergence" has many real-life applications. It can be used in fields such as finance, where infinite sums are used to calculate interest and investment returns. It can also be applied in physics and engineering, where infinite series are often used to model and solve complex problems.