Very simple question about limit superior (or upper limit)

  1. 1. The problem statement, all variables and given/known data

    If Sn =1/0! + 1/1! + 1/2! +.... 1/n! , Tn=(1+(1/n))^n, then lim sup Tn ≤ e? (e=2.71...)

    2. Relevant equations

    1. e= Ʃ(1/n!)
    2. If Sn≤Tn for n≥N, then lim sup Sn ≤ lim sup Tn

    3. The attempt at a solution

    By binomial theorem,
    Tn= 1 + 1 + 1/(2!)(1-1/n) + 1/(3!)(1-1/n)(1-2/n) + ..... + 1/n!
    Hence Tn ≤ Sn< e,
    lim sup Tn ≤ lim sup Sn< e

    ∴ lim sup Tn< e

    But I do not get lim sup Tn ≤ e

    What did i do wrong?
  2. jcsd
  3. jgens

    jgens 1,621
    Gold Member

    1. If you show that limsup Tn < e, then it follows that limsup Tn ≤ e. It turns out that it is not true that limsup Tn < e (so you did mess up in coming to that conclusion), but this conclusion does not contradict the claim that Tn ≤ e.
    2. If Tn ≤ Sn < e, then we can conclude that limsup Tn ≤ limsup Sn ≤ e. We cannot, however, conclude that limsup Sn < e based on this information; in fact, limsup Sn = e.
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