Math 1B: Calculus Spring 2020

Sample Project: e is IrrationalInstructor: Alexander Paulin Date: March 25 2020

1. Using the Taylor series for the for ex, express e as an infinite series.

For the reminder of the project let us assume that e is rational, i.e. there exist positivewhole numbers a and b, such that e = a

b .

2. Prove that b!e is an integer.

3. Prove that for n, a non-negative whole number such that n b, b!n! is an integer.

1

The theory at Taylor series tells us that de I xx ITFor all x Hence we burn thate l t l t T t T 4 IT

product ofintegersae b e a b 1 I

b Ca G c b z Z 1

This is a product of integers and is hence an integer

b Xx Xxxx x x n i x buti b

n l Xxxx XX

Hence b is an integer productuet integers

2 Project : e is Irrational

4. Prove that, under the assumption that e = a/b, the infinite series

1X

n=b+1

b!

n!

is an integer.

5. By expanding the terms of the previous infinite series, bound it above by a geometric series. Hint:

0 < c < d ) 1cd < 1

c2 .

e l t l t IT IT t a j t tis t

bi e bi bit i 3 bn I ball artergeeHence L

b lb e Lb b bz bs bn If n lu be

must be an integer

b I b l b b lt t t

ul b1111 Cb1211 Lb13u bt

I 1 Icb I 1 biz b1111621 bts

nI I It b ib ti bei bei bei bei bil

n c1

Geometric series withr

I t b

Project : e is Irrational 3

6. Using this, prove that

0 <1X

n=b+1

b!

n!< 1.

7. Prove that e is irrational.

b lo f c btl Lul b

n b11 n c lb

2 Cec 3 e is not an integer b I

b lo a ni c T I

u b11

It e a where a and b are positive wholeb

numbers then

but is an integeru b11

anda

b lO f C ln I

n b 1

This is a contradiction e is irrational