If $x\neq 0$ and if $x^2$ is rational, then $(\forall k\in\mathbb{Q}\setminus\{0,-1,-2,\ldots\}):f_k(x)\neq0\text{ and }\frac{f_{k+1}(x)}{f_k(x)}\notin\mathbb{Q}.$
If $(\exists k\in\mathbb{Q}\setminus\{0,-1,-2,\ldots\}):f_k(x)=0\text{ or }\frac{f_{k+1}(x)}{f_k(x)}\in\mathbb{Q}$
, then $x=0$ or if $x^2$ is irrational @TedShifrin
Problem:
Derive the Center of Mass of a semi-circular disk of mass $M$ and radius $R$.
$$$$
My attempt:
$$Y_{CM}=\int ydm$$
Now, $$dm=\sigma dA$$ where $\sigma$ is mass per unit area.
Converting into Cylindrical Coordinates,$$dA=rdrd\theta$$. Also, $$y=r\sin\theta$$
Hence the integral...
When I look at simple open problems in calculus ( not including analytic Number theory mainly ) most conjectures are , or are equivalent to , infinite sums or integrals being equal to a closed form real number.
So I wonder about famous open problems where we have the conjecture that an infinite...
