I am having trouble understanding why $\int_{0}^{\infty} f(t)x^tdt$ is the continuous analogue of the power series $\sum_{0}^{\infty} f(n)x^n$.
as in my mind a continuous power series is the following
(1) $f(0)x^0 + f(dt)x^{dt} + f(2dt)x^{2dt} + .... $
not the integral $\int_{0}^{\infty} f(t)x^tdt$
(2) $dt(f(0)x^0 + f(dt)x^{dt} + f(2dt)x^{2dt} + ....) $
as it seems to just be a continuous power series multiplied by $dt$
So why is (2) a "continuous analogue of a power series" but (1) isn't ?