ebx cos cx is expanded in a Taylor series ∑ an xn. b and c are positive reals. Show that either all an are non-zero, or infinitely many an are zero.
ebx cos cx = Re e(b+ic)x, so n! an = Re( (b + ic)n ). Let b + ic = k eiθ, then n! an/kn = cos nθ. So an = 0 iff nθ = (2m+1)π/2 for some integer m. So if there are any zeros then there are infinitely many.
31st Putnam 1970
© John Scholes
25 Aug 2001