let-from-R-and-u-n-2cos-u-n-1-u-n-2-0-withn-2-find-u-n-and-study-its-convrgence-

Question Number 36920 by maxmathsup by imad last updated on 07/Jun/18

l e t α f r o m R a n d u_{n} - 2 c o s (α) u_{n - 1} + u_{n - 2} = 0 w i t h n ⩾ 2

f i n d u_{n} a n d s t u d y i t s c o n v r g e n c e .

Commented by math khazana by abdo last updated on 10/Jun/18

(e) \Rightarrow u_{n + 2} - 2 c o s α u_{n + 1} + u_{n} = 0

t h e c a r a c t e r i s t i c e q u a t i o n i s

x^{2} - 2 c o s α x + 1 = 0

Δ^{^{'}} = {c o s}^{2} α - 1 = - {s i n}^{2} α = {(i s i n α)}^{2} \Rightarrow

x_{1} = c o s α + i s i n α = e^{i α}

x_{2} = c o s α - i s i n α = e^{- i α}

u_{n} = a e^{i n α} + {b e}^{- i n α}

u_{0} = a + b

u_{1} = a e^{i α} + b e^{- i α} = a e^{i α} + (u_{o} - a) e^{- i α}

= 2 i a s i n (α) + u_{0} e^{- i α} \Rightarrow 2 i a s i n α = u_{1} - u_{0} e^{- i α}

i f s i n α \neq o w e g e t a = \frac{u_{1} - u_{o} e^{- i α}}{2 i s i n α}

b = u_{0} - a = u_{0} - \frac{u_{1} - u_{0} e^{- i α}}{2 i s i n α} \dots .