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An Engel series is a sum of reciprocals of a non-decreasing sequence $(x_n)$of positive integers, which is such that each term is divisible by the previousone, and a Pierce series is an alternating sum of the reciprocals of a sequencewith the same property. Given an arbitrary rational number, we show that thereis a family of Engel series which when added to it produces a transcendentalnumber $\alpha$ whose continued fraction expansion is determined explicitly bythe corresponding sequence $(x_n)$, where the latter is generated by a certainnonlinear recurrence of second order. We also present an analogous result for arational number with a Pierce series added to or subtracted from it. In bothsituations (a rational number combined with either an Engel or a Pierceseries), the irrationality exponent is bounded below by $(3+\sqrt{5})/2$, andwe further identify infinite families of transcendental numbers $\alpha$ whoseirrationality exponent can be computed precisely. In addition, we construct thecontinued fraction expansion for an arbitrary rational number added to an Engelseries with the stronger property that $x_j^2$ divides $x_{j+1}$ for all $j$.

Continued fractions and irrationality exponents for modified Engel and Pierce series