Is an Infinite Sum of Rational Numbers Always Rational?

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Discussion Overview

The discussion revolves around the nature of infinite sums of rational numbers and whether such sums can be rational or irrational. Participants explore the implications of convergence in infinite series and the distinction between partial sums and limits.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants argue that if the sum of two rational numbers is rational, then an infinite sum of rational numbers that converges should also be rational, questioning the logic behind this assumption.
  • Others point out that while partial sums of an infinite series can be rational, the limit of the series may converge to an irrational number, indicating a distinction between the two concepts.
  • A participant highlights that certain infinite series, despite having rational terms, can converge to transcendental numbers, suggesting that intuition about sums may not align with mathematical reality.
  • Another participant emphasizes that every real number can be expressed as the sum of an infinite series of terminating decimals, which are rational, but questions the assumption that properties of finite sums apply to infinite sums.
  • Clarifications are made regarding the terminology of infinite sums and limits, with some participants reiterating that the limit of the series represents the actual sum in the context of infinite series.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between rational sums and their limits, with no consensus reached on whether an infinite sum of rational numbers can be rational or must be irrational.

Contextual Notes

The discussion includes assumptions about convergence and the nature of infinite series, which may not be universally agreed upon. The implications of limits versus actual sums are also a point of contention.

captain
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if two rational numbers added together is still rational then wouldn't an infinite sume of rational numbers that converge also be rational and if that is the case then an irrational number is therefore rational which makes no sense though. i don't see where the flaw in this lies because it is logically inconsistent.
 
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If you take a partial sum of an infinite series that converge to an irrational number then you would get a rational number. However, the point is that you never stop adding, so it tends to an irrational.
 
captain said:
if two rational numbers added together is still rational then wouldn't an infinite sume of rational numbers that converge also be rational
Why would you think that?
 
captain said:
if two rational numbers added together is still rational then wouldn't an infinite sume of rational numbers that converge also be rational

no this can easily be seen by looking at .101001000100001... this number is actually transcendental but it’s power series representation has nothing but rational terms i.e.

1/10 + 1/10^3 + 1/10^6 + 1/10^10 + 1/10^15…

Just because something intuitively seems it should be a certain way in math doesn’t mean it is. Math is about what you can deduce logically, not what you feel something should be like.
 
Every real number, rational or irrational, is the sum of an infinite number of termiinating decimals. That is, the sum of an infinite set of rational numbers.

For example, [itex]\pi[/itex]= 3+ 0.1+ 0.04+ 0.001+ 0.0005+ 0.00009+ 0.000002+ ...


Why would you think that what is true for a finite sum is necessairly true for an infinite sum?
 
the "limit" if the series is irrational, not the actual sum

infinite sum is just a simple notation of writing sum to n where n -> inf.
 
thanks for all your help i just wanted to clarify that for myself
 
leon1127 said:
the "limit" if the series is irrational, not the actual sum

infinite sum is just a simple notation of writing sum to n where n -> inf.
For an infinite series, the limit of the partial sums is the "actual sum".
 

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