E is irrational, Mathematics

If e were rational, then e = n/m for some positive integers m, n. So then 1/e = m/n. But the series expansion for 1/e is

1/e = 1 - 1/1! + 1/2! - 1/3! + ...

Call the first n values of this alternating series S(n). How good is this approximation to e? Well, the error is bounded by the next part of the alternating series:

0 < | 1/e - S(n) | = | m/n - S(n)| < 1/(n+1)!

But multiplying through by n!, you can see that

0 < | integer - integer | < 1/(n+1) < 1.

But there is no integer between 0 and 1, so this is a contradiction; e must be irrational.

Posted Date: 3/30/2013 2:49:36 AM | Location : United States

Related Discussions:- E is irrational, Assignment Help, Ask Question on E is irrational, Get Answer, Expert's Help, E is irrational Discussions

Write discussion on E is irrational
Your posts are moderated
Related Questions

if a circles diameter is 42 mm its radius is _________________ because ________________________.

hi i would like to ask you what is the answer for [-9]=[=5] grade 7

Question: Find the quotient and remainder when f(x) = x 5 - x 4 - 4x 3 + 2x + 3 is divided by g(x) = x-2. Make sure the quotient and remainder are clearly identified.

ABCD is a rhombus. the sides of the rhombus are 8cm long .one of its diagonals is 12cm .find the angels of the rhombus

Show all your work. 80% of your score is for correct justified answers; 20% is for correctly and clearly demonstrating why. For the graphing problems, use www.desmos.com/calculator

how do you convert a fraction to a percent?

Quotient Rule : If the two functions f(x) & g(x) are differentiable (that means the derivative exist) then the quotient is differentiable and,

S olve the subsequent IVP. dv/dt = 9.8 - 0.196v;               v(0) = 48 Solution To determine the solution to an Initial Value Problem we should first determine the gen