Continued fractions and irrationality exponents for modified engel and pierce series

An Engel series is a sum of reciprocals of a non-decreasing

by the previous one, and a Pierce series is an alternating sum of the

number, we show that there is a family of Engel series which when

expansion is determined explicitly by the corresponding sequence

second order. We also present an analogous result for a rational number

rational number combined with either an Engel or a Pierce series), the

identify infinite families of transcendental numbers ? whose irrationality

continued fraction expansion for an arbitrary rational number added to

j.