Question

Show that: $\sum_{n=0}^{\infty}(-1)^{n+1}\left(\frac{F_n}{F_{n+1}F_{n+2}}\right)^2=\frac{1}{\phi^3}$

score 152 · Answer 1 · 2026-03-25 18:59:06

152

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Show that: $\sum_{n=0}^{\infty}(-1)^{n+1}\left(\frac{F_n}{F_{n+1}F_{n+2}}\right)^2=\frac{1}{\phi^3}$

Published on 25 Mar 2026 - 18:59

score 54 · Answer 2 · 2020-03-29 18:02:55

54

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How to evaluate : $\sum_{n=0}^{\infty}\left(\frac{F_{n+x}}{F_{n+x+1}^2}-\frac{F_{n+x+2}}{F_{n+x+3}^2}\right)$

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score 140 · Answer 3 · 2020-03-29 19:05:28

140

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156

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score 55 · Answer 5 · 2020-04-11 13:27:44

55

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score 65 · Answer 6 · 2020-04-14 18:27:36

65

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score 29 · Answer 7 · 2020-04-15 09:44:35

29

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115

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score 69 · Answer 9 · 2020-04-17 18:37:22

69

Views

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248

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483

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score 35 · Answer 12 · 2026-03-25 18:57:43

35

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score 73 · Answer 13 · 2026-03-25 18:55:04

73

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score 147 · Answer 14 · 2020-04-25 12:06:45

147

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score 144 · Answer 15 · 2026-03-25 18:52:34

144

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List Question

Show that: $\sum_{n=0}^{\infty}(-1)^{n+1}\left(\frac{F_n}{F_{n+1}F_{n+2}}\right)^2=\frac{1}{\phi^3}$

How to evaluate : $\sum_{n=0}^{\infty}\left(\frac{F_{n+x}}{F_{n+x+1}^2}-\frac{F_{n+x+2}}{F_{n+x+3}^2}\right)$

Prove Fib(n) is closest integer to golden ratio

Radius of convergence of $\frac{z}{1-z-z^2}$

Find solutions of $f(2020)x+f(2019)y=1$ where $f$ is Fibonacci sequence

How to solve this sum of a recurrence relation?

Fibonacci numbers induction proof

Limit of the ratio of consecutive Fibonacci numbers as n approaches negative infinity?

Prove $F_{n-a+1}F_a + F_{n-a}F_{a-1}$

Why do all primes in Fibonacci numbers repeat so regularly

Fibonacci recurrence relation - Principle of Mathematical Induction

Is it natural or the way that the numerical system was invented that these magics with numbers exist in Nature?

(why) is this ratio the golden ratio?

How to prove the following strange relation concerning fibonacci numbers

A conjecture concerning Fibonacci

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