Dear all,

For anyone who wants a little puzzle to start the week: if I set z_0 = i, and z_(n+1) = 0.5 (z_n + |z_n|), what is the limit of z_n as n tends to infinity?

(Spoiler also below) . . . . . . . . . . . . . . . . . . . . . . . . . . Effectively, you are finding the arithmetic mean of z_n and |z_n| and iterating. Viewing this as a sequence of right-angled triangles, we have that the limit is the infinite product: cos(pi/4) * cos(pi/8) * cos(pi/16) * cos(pi/32) * ... Each of the cosines in the product can be converted into the exp(i*theta) malarky, yielding the following infinite product: [exp(i*pi/4) + exp(-i*pi/4)] [exp(i*pi/8) + exp(-i*pi/8)] ... If we multiply out the first four terms, we get: [exp(-15/32 * i * pi) + exp(-13/32 * i * pi) + exp(-11/32 * i * pi) + ... + exp(13/32 * i * pi) + exp(15/32 * i * pi)]/16 More generally, if we multiply the first n terms, we get the barycentre of 2^n points placed equidistantly on the unit semicircle in the positive half-plane. Since the barycentre of a uniform unit semicircle is 2/pi from the origin, our limit is indeed 2/pi. An excellent problem; I commend you. Sincerely, Adam P. Goucher