ThmDex – An index of mathematical definitions, results, and conjectures.
Riemann integral analogue to infinite geometric series
Formulation 2
Let $q \in (0, 1)$ be a D5407: Positive real number such that
(i) $f : [0, \infty) \to [0, \infty)$ is an D4367: Unsigned real function
(ii) \begin{equation} f(x) = q^x \end{equation}
Then
(1) $f$ is an D6103: Improperly Riemann integrable real function on $[0, \infty)$
(2) \begin{equation} \int^{\infty}_0 q^t \, d t = - \log q \end{equation}
Proofs
Proof 1
Let $q \in (0, 1)$ be a D5407: Positive real number such that
(i) $f : [0, \infty) \to [0, \infty)$ is an D4367: Unsigned real function
(ii) \begin{equation} f(x) = q^x \end{equation}
Let $b \in (0, \infty)$ be a positive real number. Using R3184: Second fundamental theorem of Riemann integral calculus, we have \begin{equation} \int^b_0 q^t \, d t = \left[ q^t \log q \right]^b_0 = q^b \log q - q^0 \log q = q^b \log q - \log q \end{equation} Since $q^b \to 0$ as $b \to \infty$, then \begin{equation} \int^{\infty}_0 q^t \, d t = - \log q \end{equation} $\square$