The web site BrainDen.com poses daily problems. The "poser" for today can provoke some discussion.
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...
Source: BrainDen.com (4/18/2010)
Hint: Think about partial sums of the harmonic series...
Rethink meaning of irrational, rational, and idea of a limit for a series.
Solution Commentary: Several people have posted comments to BrainDen.com's "poser":
The answer can be deciphered from these comments....except I raise a new question: Are the terms "not converge" and "diverge" equivalent (i.e. mean exactly the same thing)?
- Dun Dun Dun's reply: 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).
Oops, when I said "never diverges" I meant "diverges" (as in, never converges). haha
- mmiguel1 said: 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.
...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).