We present new algorithms for the inverse, quotient, or square root of power series. The key trick is a new algorithm -- RecursiveMiddleProduct or RMP -- computing the $n$ middle coefficients of a $2n x n$ product in essentially the same number of operations -- $K(n)$ -- than a full $n x n$ product with Karatsuba's method. This improves previous work of Mulders, Karp and Markstein, Burnikel and Ziegler. These results apply both to series, polynomials, and multiple precision floating-point numbers.