Prove an Infinite Series is Irrational

TylerH

Is it possible and is there a general method?

ramsey2879

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

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

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

converges to 2.

TylerH

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

Bacle

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

Attached Files

A theorem on irrationality of infinite series.pdf (146.5 KB, 40 views)

ramsey2879

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

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

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

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.