# real analysis – How does \$frac29 + frac{2}{9^2}+frac{2}{9^3}+cdots=frac14\$ imply that the base 3 expansion of \$1/4\$ is \$0.020202…\$?

Converting between base $$b$$ and base $$b^k$$ for $$kge2$$ is extremely simple: one base-$$b^k$$ digit corresponds one-for-one with $$k$$ base-$$b$$ digits.

The given result shows that the base-$$9$$ expansion of $$frac14$$ is $$0.overline2$$. Each digit $$2$$ becomes $$02$$ in base $$3$$, yielding the desired result $$frac14=0.overline{02}$$.