Let zₙ = cos(pi/((2n+1)(2n+3))) + i*sin(pi/((2n+1)(2n+3))). Evaluate the limit as n - infinity of the product z₁ * z₂ * z₃ *... * zₙ.

Question

Let zₙ = cos(pi/((2n+1)(2n+3))) + i*sin(pi/((2n+1)(2n+3))). Evaluate the limit as n -> infinity of the product z₁ * z₂ * z₃ *... * zₙ.

Accepted Answer

cos(pi/6) + i*sin(pi/6). The product equals cos(S) + i*sin(S) where S = sum of the arguments; the telescoping sum converges to pi/6.

Let zₙ = cos(pi/((2n+1)(2n+3))) + isin(pi/((2n+1)(2n+3))). Evaluate the limit as n -> infinity of the product z₁ z₂ * z₃ ... zₙ.