The usual power series sumalpha_{n}x^{n} are generalized by the substitution xto g(x) to power series built from a function sum a_{n}g^{n}(x). The key feature of our approach consists in using functions g with free parameters g(x)=g({p_{i}},x); the resulting expansions thus keep the Taylor—like behavior (i.e., they match derivatives at the expansion point), but can also be tuned to adjust the far—away behavior. We present six specific cases g_{X}, Xin{A,dots,F}, each containing five free parameters. We use examples to demonstrate that the generalization can improve the convergence (rate and domain), mimic branch points and branch cuts and have other interesting properties. Because the partial Bell polynomials are a necessary ingredient in our method, our results can be interpreted as new formulas for specific values of the Bell polynomials.