[Guest]

The sum of any number of rational numbers will always be rational, right?

but what about infinity?

the infinite sum of 1/n! is equal to e which is not only irrational, but transcendental as well!

so here's my question: will the harmonic series ever become irrational or will it remain rational?

prove your answer.

note: the infinite sum of 2^^-n is 1, a rational number...

Good luck!

[Guest]

Infinity is not a number

[Guest]

Calc1/Calc2/Calc3 probably Schaum's series or Apostol... "e"

http://en.wikipedia.org/wiki/Factorial is not really the "harmonic series" http://en.wikipedia.org/wiki/Harmonic_series_(mathematics), and "pi"="π" is pretty fun too

[unreality]

Validictus: he isn't saying infinity as a number, he's saying that after an infinite number of terms, sigma 1/n! becomes irrational (e)

profmmv: he's not saying that factorial and harmonic series are the same, he gave 1/factorial as an example of rational becoming irrational after an infinite number of steps, THEN asks about the harmonic series

the harmonic series can never be irrational. Like factorial, any number of steps along the way wil always be rational becauase there's always a common denomenator that can be found (highest possible n!, but can be lower than that, for example 1/1, 1/2, 1/3, 1/4, 1/5 and 1/6 can use 1/60 as a common denom). And the harmonc series never diverges, so its infinite sum can't be irrational either (like how 1/n! worked out)

[unreality]

oops, when I said *never diverges* I meant *diverges* (as in, never converges). haha

[Guest]

Validictus: he isn't saying infinity as a number, he's saying that after an infinite number of terms, sigma 1/n! becomes irrational (e)

profmmv: he's not saying that factorial and harmonic series are the same, he gave 1/factorial as an example of rational becoming irrational after an infinite number of steps, THEN asks about the harmonic series

the harmonic series can never be irrational. Like factorial, any number of steps along the way wil always be rational becauase there's always a common denomenator that can be found (highest possible n!, but can be lower than that, for example 1/1, 1/2, 1/3, 1/4, 1/5 and 1/6 can use 1/60 as a common denom). And the harmonc series never diverges, so its infinite sum can't be irrational either (like how 1/n! worked out)

I second unreality's answer.

The harmonic series diverges.

A similar series that converges is the alternating harmonic series which converges to ln(2) clearly irrational.

Another is the sum n=1 to infinity 1/n^2 which converges to pi^2/6 which is transcendental.

It is not good to assume that an infinite series is rational.

Things can get really complicated for some infinite series.

Here is something that blew my mind when I first found out about it: Riemann Series Theorem

http://en.wikipedia.org/wiki/Riemann_series_theorem

Basically if you rearrange the terms in some types of infinite series (conditionally convergent ones), then you can get different answers than you would have gotten with another arrangement.

In fact, you can get any value you want by finding a specific ordering that corresponds to that value (or you can even make the series diverge).

Neat stuff.

Edited April 18, 2010 by mmiguel1

[Guest]

Given the confines of what you've said I'd say that the harmonic series is irrational at infinity since infinity is irrational.

But my gut says that the harmonic series is never irrational. And technically 1/n! is never irrational either. It's just a rational number that can get very very close to e.

Edited April 19, 2010 by Tuckleton

[Guest]

The sum of any number of rational numbers will always be rational, right?

Right. This is always true. Therefore the sum of the first n harmonic numbers will always be rational.

In your counter example you said that the sum of 1/n! is e. In fact e is the limit of this sequence and so this does not disprove your original statement.

Anyway if you still want something more satisfactory as a proof then:

Simply take the first n harmonic numbers, combine these fractions to make a single fraction with a denominator of n!. This is a fraction and therefore rational.

if n was infinity the sum is undefined and so whether or not it is rational is irrelevant.

[Guest]

Given the confines of what you've said I'd say that the harmonic series is irrational at infinity since infinity is irrational.

But my gut says that the harmonic series is never irrational. And technically 1/n! is never irrational either. It's just a rational number that can get very very close to e.

Is infinity irrational?

If you operate in the extended complex plane (Riemann Sphere), then any non-zero number a, divided by zero gives infinity.

In such a mathematical setting, infinity can be expressed as the ratio of two integers, such as 1/0.

[Guest]

Is infinity irrational?

If you operate in the extended complex plane (Riemann Sphere), then any non-zero number a, divided by zero gives infinity.

In such a mathematical setting, infinity can be expressed as the ratio of two integers, such as 1/0.

Infinity is not a number. it is a concept. You cannot say it is "rational" or "irrational". To me that's like saying is a chair odd or even.

[Guest]

Infinity is not a number. it is a concept. You cannot say it is "rational" or "irrational". To me that's like saying is a chair odd or even.

It is an actual number in the extended complex plane.

http://en.wikipedia.org/wiki/Riemann_sphere

It's a special number that gobbles up most other numbers that add, multiply, etc.. with it.

Kind of like how zero gobbles up numbers that multiply with it.

You are right that in most contexts it is not considered to be a number, and you may be right that it is nonsensical to apply the terms rational or irrational to infinity.

In the extended complex plane, division by zero is actually defined (except for 0/0).

If you take your definition of a rational number as one that can be expressed as the quotient of two integers, then in the extended complex plane infinity is in fact rational as it can be expressed by the quotient of two integers, (e.g. 1/0).

But you are free to take whatever definition you want in whatever context you want, so you can just as easily change the rational number definition to say, except in the special case where the denominator is zero, and the quotient is irrational.

I guess the answer to the question: "is infinity rational?", is "only if you define it to be".

Edited April 21, 2010 by mmiguel1