A calculus of the absurd

3.4.3 "Hidden" geometric series.

Sometimes geometric series can be hiding in plain sight!

For example, this series is actually a geometric series! 55 This requires knowledge of logarithms and exponents, which are explored lower down in this document.

\begin{equation} \log _3\left (3^{1} \cdot 3^{\frac {1}{2}} \cdot 3^{\frac {1}{4}} \cdot ...\right ) \end{equation}

Applying the log rules, we get

\begin{equation} \log _3\left (3^1\right ) + log_3\left (3^{ \frac {1}{2} }\right ) + \log _3\left ( 3^{\frac {1}{4}}\right ) + ... \end{equation}

Which simplifies to

\begin{equation} 1 + \frac {1}{2} + \frac {1}{4} + ... \end{equation}

Which is just an infinite geometric series that converges to \(2\) (as \(\abs {r} < 1\)).