Is an Infinite Sum of Rational Numbers Always Rational?

[captain]

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.

 

[bel]

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.

 

[Hurkyl]

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?

 

[JonF]

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.

 

[HallsofIvy]

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, $\pi$= 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?

 

[leon1127]

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.

 

[captain]

thanks for all your help i just wanted to clarify that for myself

 

[HallsofIvy]

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".