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
A dealer sells a toy for Rs.24 and gains as much percent as the cost price of the toy. Find the cost price of the toy. Ans:    Let the C.P be x ∴Gain = x % ⇒ Gain = x

limit x APProaches infinity (1+1/x)x=e

The Shape of a Graph, Part II : In previous we saw how we could use the first derivative of a function to obtain some information regarding the graph of a function.  In this secti

Describe about Arithmetic and Geometric Series? When the terms of a sequence are added together instead of separated by commas, the sequence becomes a series. You will use seri


Recognizes the absolute extrema & relative extrema for the following function.                           f ( x ) = x 2       on [-1, 2] Solution:  As this function is simpl

How can i get a better understanding of logistics without having a degree on logistics and knowledge of it? Simply, in a very basic form..

A={2,3,5,7,11} B={1,3,5,7,9} C={10,20,30,40,......100} D={8,16,24,32,40} E={W,O,R,K} F={Red,Blue,Green} G={March,May} H={Jose,John,Joshua,Javier} I={3,6,9,12,15}

E1) From your experience, and what you have studied so far, by which age would-you expect an average child to be ready to acquire the following concepts? i) Simple classificatio

G raph y = sec ( x ) Solution: As with tangent we will have to avoid x's for which cosine is zero (recall that sec x =1/ cos x) Secant will not present at