Question

Cum pot rezolva aceasta limita? Idei?

Answer 1

ln(L) = lim n-> ꝏ (lim x-> 0 (ln(cosx•cos(2x)•...•cos(nx))^(1/(n³x²))

lim x -> 0 (ln(cosx•cos(2x)•...•cos(nx))/(n³x²)) =

lim x -> 0(ln(cosx)/(n³x²) + ln(cos(2x))/(n³x²)+...+ln(cos(nx))/(n³x²)) =

= lim x -> 0 (sum ln(cos(kx))/n³x²) =

= lim x -> 0 (sum (-ksin(kx))/(cos(kx)) /(2xn³)) =

= lim x -> 0 (sum (-ksin(kx) / (cos(kx)(2xn³)) ) =

= lim x -> 0 sum (-k²cos(kx)/((-ksin(kx)•(2xn³)+(cos(kx))(2n³)) =

= sum (-k²/(2n³)) = 1/(2n³)•sum(-k²) =

= -1/(2n³)•(n(n+1)(2n+1))/6 =

= -(n(n+1)(2n+1))/(12n³)

=> ln(L) = lim n -> ꝏ -(n(n+1)(2n+1))/(12n³) = lim n -> ꝏ -(2n³+.....)/(12n³) = -2/12 = -1/6 =>

=> ln(L) = -1/6 => L = e^(-1/6)