  Polynomial series approximations are a central theme in approximation theory.
Two types of series, which are featured most prominently in pure and applied
mathematics, are Taylor series expansions and expansions derived based on
families of $L^{2}-$orthogonal polynomials on bounded intervals. The identity
theorem implies that all such series agree on $\mathbb{C}$ in the limit. This
motivates an effort to derive properties, which relate the original function to
a truncated series based on $L^{2}-$orthogonal polynomial approximation which
hold on unbounded intervals. In particular, the Chebyshev series expansion of
$e^{x}$ is studied and an algebraic criterion which can be used to confirm
bounds analogous to the Taylor series upper and lower bound estimates for $x<0$
is provided. In addition to this, the generalizations to other settings
involving different functions and alternative families of $L^{2}$-orthogonal
polynomials are discussed.
