Simplify this infinite convergent sum.

Dec 2016
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Simplify the infinite convergent sum as much as possible in terms of \(\displaystyle \phi\):

\(\displaystyle \dfrac{1}{\phi^1} + \dfrac{1}{\phi^2} + \dfrac{1}{\phi^3} \ + \ ... \)


where \(\displaystyle \phi\) equals \(\displaystyle \dfrac{\sqrt{5} + 1}{2}\).





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Source:

The Call Of The Primes: Surprising Patterns, Peculiar Puzzles, and Other Marvels of Mathematics

by Owen O'Shea

pages 123, 125-126
 

SlipEternal

MHF Helper
Nov 2010
3,728
1,571
This is a geometric series:

$\sum_{n\ge 1}\left(\dfrac{1}{\phi}\right)^n = \left[\sum_{n\ge 0}\left(\dfrac{1}{\phi}\right)^n\right]-1=\dfrac{1}{1-\dfrac{1}{\phi}}-1 = \dfrac{1}{\phi-1}=\phi $
 
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