The transcendental number e – Euler’s number(adsbygoogle = window.adsbygoogle || []).push({});

(the underscores(_) represent subscript and (^) represent superscript/exponet)

The limiting value of sequence {e_n} where e_n = (1+1/n)^n is the irrational number e. My text gives a challenge to see if you can prove that this converges by verifying the following: (1+1/n) ^n < e < (1+1/n) ^(n+1) and that 2 ≤ e < 3 for all n Є N. There has been a consensus in the class that there is a typo in the text and that it should read: (1+1/n) ^n < e_(n+1) < (1+1/n)^( n+1) 2 ≤ e_m < 3. I’m not sure if I know enough to question this but I will accept it and proceed with the hints on how to do this.

The first hint is to

Use the binominal theorem to write out n+1 terms of e_n using the fact that n(n-1)(n-2)/n^3 can be expressed as (1-1/n)(1-2/n). then do the same for n+2 in terms of e_(n+1) such that you can come to the conclusion that e_n < e_(n+1) right off the bat in this multi step problem I’m stumped am I using the binominal expansion correctly?. So far I’ve come up with:

e_n = 1 + 1 + (1/2)(n)(n-1)(1/n^2) + (1/3)(n)(n-1)(n-2)(1/n^3) + …. + (1/n)^n

e_n = 2 + (1/2)(n)(n-1)(1/n^2) + (1/3)(1-(1/n))(1-(2/n)) + …. + (1/n)^n

then for n+1 in terms of e_n

= 2 + (1/2)(n+1)(n)(1/(n+1)^2) + (1/3)(1-(1/(n+1))(1-2/(n+1)) + …. + (1/(n+1))^(n+1)

Then for n+2 in terms of e_(n+1) here I substituted n+2 for n in the equation above, is this correct according to the given hints?

= 2 + (1/2)(n+3)(n+2)(1/(n+3)^2) + (1/3)(1-(1/(n+3))(1-2/(n+3)) + …. + (1/(n+3))^(n+3)

I wanted to get some help on these preliminary steps before I move onto the second half of the challenge. If I am correct so far how does this come to the conclusion that e_n < e_(n+1)?

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# Homework Help: Euler's number e, proving convergence and bounds

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