Prove an Infinite Series is Irrational

[TylerH]

Is it possible and is there a general method?

 

[ramsey2879]

Is it possible and is there a general method?

Don't you mean infinite non-repeating series, since infinite series, e.g. 12.66123123123..., where the ending part, i.e. 123, repeats forever are rational numbers. Prove that all rational numbers are either finite decimal numbers or infinite repeating decimals and your theorem as amended is proved.

 

[TylerH]

I mean any series. Not necessarily of the form digit * 10 ^ position.

Like [tex]\sum_{k=0}^{\infty}{\frac{1}{k!}}[/tex] for example, which is e, which is irrational, but can I prove that based solely on the series?

 

[ramsey2879]

I mean any series. Not necessarily of the form digit * 10 ^ position.

Like [tex]\sum_{k=0}^{\infty}{\frac{1}{k!}}[/tex] for example, which is e, which is irrational, but can I prove that based solely on the series?

Whatever the form of the infinite series, I just showed that some infinite series are not irrational. So there is a counterexample to your claim. That is you can't demonstrate that a series is irrational simply by the fact that it is an infinite series.

 

[TylerH]

Oh, I see what you mean. There is a misunderstanding. (Likely due to my ambiguous wording.) I wasn't referring to all infinite series, but the concept of a single, general instance. More formally stated: Given an infinite series, assuming it is convergent, is there a way to prove that the number it converges to is irrational?

 

[disregardthat]

For sequences with rational terms, it might be easier. This is the case for ln 2 = 1-1/2+1/3-1/4+... and e = 1/0!+1/1!+1/2! + ... Assume it is on the form a/b, and simply use the denominators of the terms to find a contradiction. This is a classical way of proving irrationality of e. It might not always work, but it's worth a try.

The euler-mascheroni constant [tex]\gamma = \lim_{n \rightarrow \infty } \left( \sum_{k=1}^n \frac{1}{k} - \ln(n) \right) = 1 + \sum_{k=2}^{\infty} \frac{1}{k} - (\ln(k) - \ln(k-1))[/tex] is not known to be rational nor irrational yet.

 

[lurflurf]

There is no general method. The series for e though is a standard easy example.
Consider
1/(N+1)<N!e-[N!e]<1/N
for any large integer N where [] denotes the floor
-><-

 

[Bacle]

Oh, I see what you mean. There is a misunderstanding. (Likely due to my ambiguous wording.) I wasn't referring to all infinite series, but the concept of a single, general instance. More formally stated: Given an infinite series, assuming it is convergent, is there a way to prove that the number it converges to is irrational?

Am I missing something? 1+1/2+...+1/2ⁿ+...

converges to 2.

 

[TylerH]

Am I missing something? 1+1/2+...+1/2ⁿ+...

converges to 2.

It does... I don't see the connection.

 

[Bacle]

Oh, I see what you mean. There is a misunderstanding. (Likely due to my ambiguous wording.) I wasn't referring to all infinite series, but the concept of a single, general instance. More formally stated: Given an infinite series, assuming it is convergent, is there a way to prove that the number it converges to is irrational?

Well, 1+1/2+...+1/2^n+... is an infinite series that converges, and converges
to a rational number, which, if I understood you well, is a counterexample
to your claim that every convergent infinite series converges to an irrational number.

Did I misunderstand ( or misunderestimate :) ) your question?

 

[TylerH]

I made no claims(notice the lack of a period in my restatement.). I asked whether there is a general method to tell if a convergent series converges to an irrational number or not(notice the question mark.).

 

[myth_kill]

see the attached document for a comprehensive take on the op along iwth some excellent sources

 

[ramsey2879]

see the attached document for a comprehensive take on the op along iwth some excellent sources

Thanks for the link. As far as I can tell this document is an answer to the question of whether for certain types of sequences that converge to an irrational number, there is a general way to prove that the sequence is irrational. This basically answers the op's question, but I cannot tell whether this method will work for all infinite sequences that converge to an irrational number.

 

[TylerH]

Yeah, that pretty well does it. Thanks.

 

[brydustin]

Am I missing something? 1+1/2+...+1/2ⁿ+...

converges to 2.

Also, 0/1 + 0/2 + 0/3 + ... = 0
these are silly examples, but they get the point across well that an infinite series can be rational.

 

[TylerH]

Also, 0/1 + 0/2 + 0/3 + ... = 0
these are silly examples, but they get the point across well that an infinite series can be rational.

Really? How many times can the same statement be misconstrued?

For the umpteenth time: I was asking if there is a way to tell if a series is irrational. NOT saying all are.

 

[ramsey2879]

Also, 0/1 + 0/2 + 0/3 + ... = 0
these are silly examples, but they get the point across well that an infinite series can be rational.

I also want to add that when I said "all infinite sequences that converge to an irrational number" I meant --all those infinite sequences that converge to an infinite number-- if that is what the confusion was about. Still not certain if the general method is applicable to all such sequences.

 

[brydustin]

Really? How many times can the same statement be misconstrued?

For the umpteenth time: I was asking if there is a way to tell if a series is irrational. NOT saying all are.

As a mathematician, I need your statements to be precise, and as a graduate student I haven't got time to read a full blog and point out your mistakes. Perhaps you should learn to use the edit function, it would do you and everyone here a real favor. Thanks and have a pleasant day. :)

 

[Char. Limit]

Wow. Brydustin and Bacle, he CLEARLY meant "Given a convergent series, is there a way to tell if the number it converges to is irrational". He did NOT mean "All convergent series are irrational." Honestly, it was obvious to even a cursory reading.

 

[TylerH]

As a mathematician, I need your statements to be precise, and as a graduate student I haven't got time to read a full blog and point out your mistakes. Perhaps you should learn to use the edit function, it would do you and everyone here a real favor. Thanks and have a pleasant day. :)

That's a good idea. I'll do that. :)

EDIT: Unfortunately, I can't edit posts as old as the OP of this thread. (Although, I do see the use of such a restriction.)

 

[Bacle]

My apologies, I did not read the statement carefully.

A point that is interesting, I think, tho, is that an infinite sum of rationals, like those in the series
for Euler's e, is irrational in the limit. So Rationals are closed under finite, but not infinite addition,
or maybe more accurately, under taking of limits.

 

[camilus]

I mean any series. Not necessarily of the form digit * 10 ^ position.

Like [tex]\sum_{k=0}^{\infty}{\frac{1}{k!}}[/tex] for example, which is e, which is irrational, but can I prove that based solely on the series?

using that definition of e, it is possible to prove irrationality.