The Student Room Group
Taylor's Theorem
Reply 3
Try evaluating

lim(n-->inf)[1+1/n]^n
shaolinbrothers
Hi, I got stuck on a project which is to prove that e = 2.71828... Could anyone help me to find the solution? I would deeply appreciate. :smile: :smile: :wink: :wink: :wink:

This very much depends how you've defined e. I would use Maclaurin's expansion (as suggested by UnspokenTruth), but unless you've already said what e is, it's difficult to do that. If you define d/dx(e^x) = e^x, then Maclaurin's expansion would be something worth looking into. This (after a bit of work) defines e as:
e = r=01r!\displaystyle \sum_{r=0}^\infty \frac{1}{r!} = 1 + 1 + 1/2! + 1/3! + 1/4! + 1/5! + ...
where n! = n*(n-1)*(n-2)...*2*1.
I would point out, since the thread title seems to suggest such thinking, that e is not 2.718281828....., with the '1828' section forever repeating. That would imply e was not only algebraic, but rational (and equal to 271801/99990), which is categorially is not. e's decimal expansion does not repeat over any finite length, since it is irrational.
if some number repaeats after decimal , it can also be irrational . yes e's value is incorrect but the fact you said is wrong
Original post by ayushman1024
if some number repaeats after decimal , it can also be irrational . yes e's value is incorrect but the fact you said is wrong


Here's a proof from "A Concise Introduction to Pure Mathematics" to show that if "some number repeats after the decimal", it MUST be rational.
your responses are over 10 years late...

Original post by I hate maths
Here's a proof from "A Concise Introduction to Pure Mathematics" to show that if "some number repeats after the decimal", it MUST be rational.


Original post by ayushman1024
if some number repaeats after decimal , it can also be irrational . yes e's value is incorrect but the fact you said is wrong
Original post by yesmynameis
your responses are over 10 years late...


Hahaha oh **** I didn't read the date. I was looking up some maths and came across this.