- #1
TylerH
- 729
- 0
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 said:Is it possible and is there a general method?
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 said: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?
TylerH said: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?
It does... I don't see the connection.Bacle said:Am I missing something? 1+1/2+...+1/2<sup>n</sup>+...
converges to 2.
TylerH said: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?
myth_kill said:see the attached document for a comprehensive take on the op along iwth some excellent sources
Bacle said:Am I missing something? 1+1/2+...+1/2<sup>n</sup>+...
converges to 2.
Really? How many times can the same statement be misconstrued?brydustin said: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 said: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 said: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.
That's a good idea. I'll do that. :)brydustin said: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. :)
TylerH said: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?
An infinite series is considered irrational if it cannot be expressed as a finite ratio of two integers. In other words, the sum of the terms in the series cannot be written as a fraction in the form of a/b, where a and b are integers.
To prove that an infinite series is irrational, one must show that it cannot be expressed as a finite ratio of two integers. This can be done through various methods such as the rational root theorem, the irrationality of certain mathematical constants, or by using the properties of infinite series.
One well-known example is the infinite series for pi, which is 3.1415926535... Other examples include the infinite series for e (2.7182818284...) and the infinite series for the golden ratio (1.6180339887...).
Proving that an infinite series is irrational is important in mathematics as it helps to understand the nature of certain mathematical constants and the properties of infinite series. It also has applications in fields such as number theory, calculus, and cryptography.
Yes, an infinite series can be both irrational and convergent. The irrationality of a series refers to the nature of its terms, while convergence refers to the sum of those terms. For example, the series 1 + 1/2 + 1/4 + 1/8 + ... is both irrational (as it approaches 2) and convergent (as the sum of its terms approaches 2).