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

Ask an Expert

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

Write your message here..

Related Questions

Polynomials, give an example of a binomial of degree 27? give an example of a binomial of degree 27?

Find the value a2 + ß2 and (a - ß)2, If  α,β are the zeros of the polynom... If  α,β are the zeros of the polynomial 2x 2 - 4x + 5 find the value of a) α 2 + β 2   b) (α - β) 2 . Ans : p (x) = 2 x 2 - 4 x + 5           (Ans: a) -1 , b) -6) α + β =

Explain identifying conic sections, Explain Identifying Conic Sections ... Explain Identifying Conic Sections The graph of a quadratic equation in the variables x and y, like this one, x 2 + 3y 2 + 6y = -4, is a conic sections. There are three kind

Find the derivatives, Find the derivatives of the following functions a)... Find the derivatives of the following functions a)      y = 5x 4 +3x -1-x 3 b)      y = (x+1) -1/2 c)      y= e x2+1 d)     y= e 3x lnx e)     y =ln(x+1/x)y

How did they go about "modernizing" the region, What were the two main poli... What were the two main political parties that formed in the majority of the new nations of Latin America post independence? In what ways were they different? Which party ascended t

Converting., I need help converting my project fractions into 1 I need help converting my project fractions into 1

Use the definition of the right- and left-handed limits, Use the definition... Use the definition of the limit to prove the given limit. Solution Let ε> 0 is any number then we have to find a number δ > 0 so that the following will be true. |

Fraction, 2 of 10 = 2 of 10 =

Define symmetric, Define symmetric, asymmetric and antisymmetric relations.... Define symmetric, asymmetric and antisymmetric relations.    Ans: Symmetric Relation A relation R illustrated on a set A is said to be a symmetric relation if for any x,

Calculate overhead in bit and time-synchronous communication, 2.    Suppose... 2.    Suppose a file of 35,000 characters is to be sent over a line at 55,000bps. 1. Calculate the overhead in bits and time using asynchronous transmission. Assume 1 start bit