Proof that e is irrational using Taylor series

Homework Statement

Using the equality ##e = \sum_{k=0}^n \frac{1}{k!} + e^\theta \frac{1}{(n+1)!}## with ##0< \theta < 1##, show the inequality ##0 < n!e-a_n<\frac{e}{n+1}## where ##a_n## is a natural number.
Use this to show that ##e## is irrational.
(Hint: set ##e=p/q## and ##n=q##)

N/A

The Attempt at a Solution

I think I manage to do the first part. Multiply both sides with ##n!##
##n!e = \sum_{k=0}^n \frac{n!}{k!} + \frac{e^\theta}{n+1}##
Setting ##\theta = 0## and ##\theta=1## respectively we have the inequality
##0 < n!e-a_n<\frac{e}{n+1}## with ##a_n = \frac{n!}{k!}##.

For the second part assume ##e= \frac{p}{q}## where ##p,q## are natural numbers without a common divisor (except 1).
Set ##n=q## in the above inequality
##0 < q!\frac{p}{q}-a_q < \frac{p}{q(q+1)}##
Since ##q,p## and ##q+1## is positive we can multiply (divide) by these
##0 < (q+1)(q!-qa_q/p)< 1##
My hopes at this point would be to have an integer in the middle and hence get a contradiction (since there is no integers between 0 and 1), but that ##p## is in the way.

Samy_A
Homework Helper

Homework Statement

Using the equality ##e = \sum_{k=0}^n \frac{1}{k!} + e^\theta \frac{1}{(n+1)!}## with ##0< \theta < 1##, show the inequality ##0 < n!e-a_n<\frac{e}{n+1}## where ##a_n## is a natural number.
Use this to show that ##e## is irrational.
(Hint: set ##e=p/q## and ##n=q##)

N/A

The Attempt at a Solution

I think I manage to do the first part. Multiply both sides with ##n!##
##n!e = \sum_{k=0}^n \frac{n!}{k!} + \frac{e^\theta}{n+1}##
Setting ##\theta = 0## and ##\theta=1## respectively we have the inequality
##0 < n!e-a_n<\frac{e}{n+1}## with ##a_n = \frac{n!}{k!}##.

For the second part assume ##e= \frac{p}{q}## where ##p,q## are natural numbers without a common divisor (except 1).
Set ##n=q## in the above inequality
##0 < q!\frac{p}{q}-a_q < \frac{p}{q(q+1)}##
Since ##q,p## and ##q+1## is positive we can multiply (divide) by these
##0 < (q+1)(q!-qa_q/p)< 1##
My hopes at this point would be to have an integer in the middle and hence get a contradiction (since there is no integers between 0 and 1), but that ##p## is in the way.
From ##0 < q!\frac{p}{q}-a_q < \frac{p}{q(q+1)}##, you get
##0<p(q-1)!-a_q<\frac{e}{q+1}##.
I you could show that ##\frac{e}{q+1}<1##, you are done.

• Incand
If you could show that ##\frac{e}{q+1}<1##, you are done.

Thanks! We know that ##2 < e < 3## (this is easy to prove using the limit definition of e). If ##q \ge 2## then the inequality is obviously true since ##e < 3## and hence we have a contradiction.

The case for ##q=1## means we have a natural number so it's enough to show it for ##q \ge 2## as above.

• Samy_A