Peculiar Sums

Calculus Level 3

$\displaystyle A = \sum_{n=1}^\infty \left ( \dfrac{1}{\phi}\right )^{n} , \qquad B= \displaystyle \sum_{n=0}^\infty \left( \dfrac{1}{\phi^{2}} \right)^{n}$

Let $\phi$ denote the golden ratio, $\phi = \frac{1+\sqrt5}2$. Then find $A+B$ to 3 decimal places.

Hint: $\frac 1\phi = \phi - 1$.

×