Saturday, 28 February 2015

logarithms - Prove that 5/2<e<3?




Prove that 52<e<3




By the definition of log and exp,
1=log(e)=e11tdt

where e=exp(1).



So given that e is unknown, how could I prove this problem? I think I'm missing some important facts that could somehow help me rewrite e in some form of 3 and 5/2. Any idea would be greatly appreciated.


Answer



e=lim



the rth term $t_r=\frac{n(n-1)(n-2)\cdots(n-r+1)}{r!n^r}=\frac1{r!}\prod_{0\le s

So, \lim_{n\to \infty}t_r=\frac1{r!}




So, e=1+\frac1{1!}+\frac1{2!}+\frac1{3!}+\cdots



But 1+\frac1{1!}+\frac1{2!}+\frac1{3!}+\cdots>1+1+0.5=2.5



Again,



3!=1.2.3>1.2.2=2^2



4!=1.2.3.4>1.2.2.2=2^3




So,



e=1+\frac1{1!}+\frac1{2!}+\frac1{3!}+\cdots
<1+1+\frac12+\frac1{2^2}+\frac1{2^3}+\cdots
=1+(1+\frac12+\frac1{2^2}+\frac1{2^3}+\cdots)=1+\frac{1}{1-\frac12}=3
as the terms inside parenthesis forms an infinite geometric series with the common ratio =\frac12, the 1st term being 1


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