lim-n-n-1-n-1-n-n-JS-

Question Number 101822 by john santu last updated on 04/Jul/20

\lim_{n \to \infty} \frac{ϕ^{n + 1} - {(- ϕ)}^{- n - 1}}{ϕ^{n} - {(- ϕ)}^{- n}} =

(J S ⊛)

Answered by bobhans last updated on 05/Jul/20

ϕ = \frac{\sqrt{5} + 1}{2} > 1; \lim_{n \to \infty} {(- ϕ)}^{- n} = \lim_{n \to \infty} \frac{1}{{(- ϕ)}^{n}} = 0

T h e n \lim_{n \to \infty} \frac{ϕ^{n + 1} - {(- ϕ)}^{- n - 1}}{ϕ^{n} - {(- ϕ)}^{- n}} = \lim_{n \to \infty} \frac{ϕ^{n + 1}}{ϕ^{n}} = ϕ

= \frac{\sqrt{5} + 1}{2} (B o b -)

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