I really like to use paradoxes in my math classes, in order to awaken the interest of my students. Concretely, these last weeks I am proposing paradoxes that achieve the conclusion that 2=1. After one week, I explain the solution in the blackboard and I propose a new one. For example, I posted the following one some months ago: What is wrong with the sum of these two series?
I would like to increase my repertoire of fake-proofs. I would be glad to read your proposals and discuss them! My students are 18 years old, so don't be too cruel :) Here is my own contribution:
y(x)=tanx
y′=1cos2x
y′′=2sinxcos3x
This can be rewritten as:
y′′=2sinxcos3x=2sinxcosx⋅cos2x=2tanx⋅1cos2x=2yy′=(y2)′
Integrating both sides of the equation y′′=(y2)′:
y′=y2
And therefore
1cos2x=tan2x
Now, evalueting this equation at x=π/4
1(√2/2)2=12
2=1
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