We present new algorithms for the inverse, division, and square root of power series. The key trick is a new algorithm --- MiddleProduct or, for short, MP --- computing the n middle coefficients of a (2n-1)*n full product in the same number of multiplications as a full n*n product. This improves previous work of Brent, Mulders, Karp and Markstein, Burnikel and Ziegler. A forthcoming paper will study the floating-point case.