|May6-12, 09:47 PM||#1|
very simple question about limit superior (or upper limit)
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?
|May6-12, 10:47 PM||#2|
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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