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Very simple question about limit superior (or upper limit) 
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#1
May612, 09:47 PM

P: 23

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!)(11/n) + 1/(3!)(11/n)(12/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
May612, 10:47 PM

P: 1,622

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


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