1. Joined
    06 Mar '12
    Moves
    625
    28 Oct '14 08:382 edits
    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:

    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. Joined
    06 Mar '12
    Moves
    625
    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. Standard memberDeepThought
    Losing the Thread
    Cosmopolis
    Joined
    27 Oct '04
    Moves
    78633
    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. Joined
    06 Mar '12
    Moves
    625
    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. Cape Town
    Joined
    14 Apr '05
    Moves
    52945
    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. Joined
    06 Mar '12
    Moves
    625
    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,"
    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?
  7. Cape Town
    Joined
    14 Apr '05
    Moves
    52945
    31 Oct '14 10:35
    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.
Back to Top