I have already asked this question to two experts at:
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:

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?

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?"

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.

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:

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.

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,"
which contradicted that.
But then the thought occurred to me that "no limit" doesn't necessarily imply "infinity"! Because "no limit" in this context means, as I think he implied by, "No matter what value is given to n, n+1 is greater.", you cannot define any specific finite limit because there is none but that doesn't logically imply n can be let alone is infinite!
-am I thinking about that in the right way?

And does anyone here disagree with him that n cannot literally be infinite?

Originally posted by humy 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.

That is true for 1/2+1/4+1/8+........ which has the finite limit 1.
It is better to say that for any finite positive real number x we can always find an n for which the partial sum of the series is greater than x.