What is wrong with this proof that $\pi=\infty$

Question

Does this "proof" show that $\pi =\infty$???

http://www.academia.edu/1611664/Sum_of_an_Infinite_sequence_PAPER

Is there something wrong with the subbing in of infinity at the end of the paper?

Answer 1

You are precisely right. The paper replaces taking the limit of an expression as $k\to \infty$ with simply plugging in the "value" $k=\infty$, which is absurd. The author then concludes that $$\pi=\infty\sin(\pi/\infty)\cdot\frac12=\infty$$ and even if the first equality where valid and the rules of extended arithmetic used to evaluate $\infty\sin(\pi/\infty)\cdot \frac12$, we get $$\infty\sin(\pi/\infty)\cdot \frac12=\infty\sin(0)\cdot\frac12=\infty\cdot 0$$ which is an indeterminate form.

Answer 2

That proof is wrong from the start.

• First off, when working with Calculus (i.e differentiation, and integration), we always use radians instead of degrees, i.e, the line: $\sin \left( \frac{180}{k} \right) = \frac{180}{k} - \frac{ \left( \frac{180}{k} \right) ^ 3}{3} + ...$ is plain wrong!!! This is because one of the fundamental/basic limits $\lim\limits_{x \to 0} \frac{\sin x}{x} = 1$ only holds when x is measured in radian.

• Secondly, when evaluating limits, one cannot just plug $\infty$ in like that, this is what others have pointed out.