## Footnote

5Rigorously: Over a period beginning at a particular time $$t = t_0$$ and ending at $$t = t_0 + T$$, the term $$e^{kt}$$ decreases from $$e^{k t_0}$$ to $$e^{k(t_0 + T)} = e^{k t_0} e^{kT} = \dfrac{1}{2} e^{kt_0}$$, that is, decreases by half.