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In this paper we discuss mode conversion in a straight fiber which maintains its circular cross section but whose radius increases monotonically with the length coordinate so that it passes from the single-mode to the multimode regime. Our principal concern is with step-index fibers which are treated in the scalar approximation. We study conversion of power from the incident dominant mode to other core and cladding modes for several functions . The results of our calculations show clearly that the dominant mode is able to adapt itself to the changing fiber radius provided the change is gradual. For comparison, we also consider a fiber with an infinitely extended radial parabolic-index profile as an approximation to graded index fibers. We show that it too exhibits mode conversion similar to that of the step-index fiber, provided that the comparison is made on the basis of corresponding mode spot sizes instead of actual core radii.