Bathroom singing, also known as singing in the bathroom, singing in the bath, or singing in the shower is a widespread phenomenon.
Many people sing in the bathroom because the hard wall surfaces, often tiles or wooden panels, and lack of soft furnishings, create an aurally pleasing acoustic environment. The multiple reflections from walls enrich the sound of one's voice. Small dimensions and hard surfaces of a typical bathroom produce various kinds of standing waves, reverberation and echoes, giving the voice "fullness and depth."
This habit was reported (with an attempt of explanations) centuries...
Let $f$ be a continuous, decreasing function, with $\displaystyle\lim_{x\rightarrow\infty}f(x)=0$.
Let $\gamma_k(x)=xf((k+1)x)-\int_{(k+1)x}^{(k+2)x}f(t)\mathrm{d}t,\displaystyle x>0$.
Let $\Gamma(x)=\sum\limits_{k=0}^{\infty}\gamma_k(x)$, suppose that $\displaystyle\lim_{x\rightarrow0} xf(x)=A...
