- 28 Oct '14 08:38 / 2 editsI 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:

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? - 28 Oct '14 14:26 / 2 editsIt 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 - 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) = infinityIt 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 - 29 Oct '14 07:46

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.*Originally posted by humy***I mean, how does one work out the limit to such a sequence? Is there a general method?** - 31 Oct '14 10:27 / 2 editsI 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? - 31 Oct '14 10:35

That is true for 1/2+1/4+1/8+........ which has the finite limit 1.*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.

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.