Find the Fallacies

Here are some examples of false Mathematical proofs:
Number 1:

Number 2:

Number 3 (the famous one):
let:

Number 4:
(1-1) + (1-1) + … + (1-1) + (1-1) + … = 0
1 + (-1+1) + (-1+1) + … + (-1+1) + … = 0
1 + 0 + 0 +… + 0 + .. = 0
1 = 0

 

A Simple Exercise in Laplace Transform

Prove that:

Solution:
First,

It follows that:

Then,

Hence,