can a circle of infinite radius be possible??

Question

According to Ramanujan's infinity theorem, the infinity is -1/12. Then if we consider a circle of this -1/12 radius it has a negative area which makes no sense. But it is said that a circle with infinite radius is a straight infinite line???? How and what to believe??

Answer 1

No it doesn't have negative area. The set of all circles with radius $r\in\Bbb R$, let's denote this $C_r$, is given by: $$C_r=\{(x-a)^2+(y-b)^2=r^2|a,b\in\Bbb R\}$$

The area of any such circle is $\pi r^2$.

Its clear to see that $$\forall r\in \Bbb R; C_r\equiv C_{-r}$$

The greater problem with your question though is that $\infty=-\frac{1}{12}$ is not correct in any sense. It is the value of $\zeta(-1)$, but this isn't the same thing!