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 the value of \(A+B\).

Give your answer to 3 decimal places.

\(\)
Hint: \(\frac 1\phi = \phi - 1 \).

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