In this paper, we study the harmonic continued fractions. These form an infinite family of ordinary continued fractions with coefficients t1,t2,t3,… for all t> 0. We derive explicit formulas for the numerator and the denominator of the convergents. In particular, when t is an even positive integer, we derive the limit value of the harmonic continued fraction. En route, we define and study convolution alternating power sums and prove some identities involving Euler polynomials and Stirling numbers, which are of independent interest.