# maths questions about a mathamatical limit

humy
Science 28 Oct '14 08:38
1. 28 Oct '14 08:382 edits
http://www.allexperts.com/el/Number-Theory/
but they haven't responded at all for ages and I have got fed up with waiting so I think I try my luck here:

What is the limit of this sequence as n tends towards infinity:

1/2 + 1/3 + 1/4 + 1/5 + …. + 1/(n-3) + 1/(n-2) + 1/(n-1) + 1/n

Is the limit +infinity? (I hope not! Not for what I want it for! ) Or is it finite?
And why is it whatever value it is? -I mean, how does one work out the limit to such a sequence? Is there a general method?

Also: does it ONLY make sense to talk about a mathematical limit like this one “as n tends towards infinity” or can you also rationally talk about a mathematical limit like this one where n is LITERALLY equal to +infinity?
2. 28 Oct '14 10:05
Just noticed I asked "Is the limit +infinity?" which doesn't makes sense because it being infinity means it has no limit by definition. That question should have been:

""Is there no limit because it tends towards infinity?"
3. DeepThought
28 Oct '14 14:262 edits
It is infinite. It is called the harmonic series, which has a good Wikipedia page. The generalised sum is:

Sum_{n € N} n^{-s} = zeta(s) ; where zeta is the Riemann Zeta function, which also has a good page on Wikipedia. (€ is the closest symbol I can produce to the one used for "is in" in set theory, N is the natural numbers)

The zeta function is known for some values. zeta(-1) = -1/12, so we get:

1 + 2 + 3 + ... = -1/12, as noted by twhitehead in the other thread.

1 + 1/2² + 1/3² + ... = zeta (2) = pi²/6

1 + 1/2 + 1/3 + ... = zeta(1) = infinity
4. 28 Oct '14 18:01
Originally posted by DeepThought
It is infinite. It is called the harmonic series, which has a good Wikipedia page. The generalised sum is:

Sum_{n € N} n^{-s} = zeta(s) ; where zeta is the Riemann Zeta function, which also has a good page on Wikipedia. (€ is the closest symbol I can produce to the one used for "is in" in set theory, N is the natural numbers)

The zeta function ...[text shortened]... thread.

1 + 1/2² + 1/3² + ... = zeta (2) = pi²/6

1 + 1/2 + 1/3 + ... = zeta(1) = infinity
It is called the harmonic series

Thanks for that.
From that I found some relevant websites to mull over starting with:

http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29

-which gives an ingenious simple proof that it is equal to infinity.

I better also study:

http://en.wikipedia.org/wiki/Riemann_zeta_function
5. 29 Oct '14 07:46
Originally posted by humy
I mean, how does one work out the limit to such a sequence? Is there a general method?
For a continuously increasing series a typical proof would be to show that for an arbitrarily large real number x you can always find an n for which the partial sum S(n) is larger than x.
6. 31 Oct '14 10:272 edits
I have finally got an answer from http://www.allexperts.com/el/Number-Theory/ to my OP question from an expert ( Scott A Wilson ) . The answer he gave was:

"...That is an infinite sum, so there is no limit.
The individual terms go to 0, but the sum does not.
Approximating this with an integral gives the integral of 1/x for x going from 1 to infinity.
Since this is ln(x), the ln() of infinity is infinity, so there is no limit.

The value of n can never be infinity, but it only tends towards infinity.
No matter what value is given to n, n+1 is greater.
That is why it is said to be the limit as n tends to infinity.
..."

At first, I thought that sounded contradictory because the "no limit" and "infinite" parts in the
"That is an infinite sum, so there is no limit."
made it sound to me that n CAN be infinite (because how can it be an "infinite sum" if n is finite? ) , but then he said;
"The value of n can never be infinity,"