20 May 2016
A couple of years ago, I came across a picture of a mathematical proof that showed that the constant Pi is equal to 4. The remarkable thing was that the error in the reasoning wasn't obvious at all, even after taking university level calculus lessons, it stumped me for a moment. The false proof goes as follows:
This is why I wrote a mini paper about this problem, which you can find here.
Recently, when I was looking into this problem again, I found another problem that is analogous. Suppose there is a helicopter that needs a certain lift force to stay in the air. Now, if we increase the length of the main rotor blades, they have to rotate at a lower angular speed to exert the same lift force. If we continue making the rotor blades larger and keep the lift force the same, the angular speed of the blades goes to zero. Therefore, so goes the erroneous conclusion, if the rotor blades have an infinite length, they do not have to turn at all and our helicopter can stay in the air indefinitely. The conclusion is that a property may hold for all functions or numbers of a convergent series, but it may not hold for the limit. And to conclude, without further proof, that it holds for the limit is a mathematical mistake.