# A calculus of the absurd

##### 3.4.3 "Hidden" geometric series.

Sometimes geometric series can be hiding in plain sight!

For example, this series (note: logarithms and exponents are explored in Section 8) is actually a geometric series!

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

Applying the log rules, we get

$$\log _3\left (3^1\right ) + \log _3\left (3^{ \frac {1}{2} }\right ) + \log _3\left ( 3^{\frac {1}{4}}\right ) + ...$$

Which simplifies to

$$1 + \frac {1}{2} + \frac {1}{4} + ...$$

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