wykaż że ... matura 1999 :P

[Matka Chrzestna]

Wykaż, że
\(\displaystyle{ 1 3^{\frac{1}{3}} 9^{\frac {1}{9}} 27^{\frac {1}{27}} ... = \sqrt[4]{27}}\)

[PFloyd]

\(\displaystyle{ =3^{\frac{1}{3}}\cdot (3^2)^{\frac{1}{3^2}}\cdot (3^3)^{\frac{1}{3^3}\cdot...}=3^{\frac{1}{3}+2\cdot \frac{1}{3^2}+3\cdot \frac{1}{3^3}...}=3^{\frac{\frac{1}{3}}{1-\frac{1}{3}}+\frac{\frac{1}{3^2}}{1-\frac{1}{3}}+\frac{\frac{1}{3^3}}{1-\frac{1}{3}}+...}=3^{\frac{1}{2}+\frac{1}{6}+\frac{1}{18}+...}=3^{\frac{\frac{1}{2}}{1-\frac{1}{3}}}=3^{\frac{3}{4}}}\)