atomthick

There is nothing odd here. The series 1+2+4+8+16+... clearly diverges but you can rewrite it in the folowing way Sum(1 + 1/(2k)^x), k=1..inf. Taking x=-1 will get you to the sum 1+2+4+6+... This function clearly diverge for x <= 1 but by analytic continuation for x < 1 you can get a more general function which has the same values as our Sum. The trick with multiplying the sum with (2-1) has enabled you to find the value of the more general function i.e. for the analytic continuation.