  In this paper we study sequences, series, power series and uniform
convergence in the $\mathcal{A}$-Calculus. Here $\mathcal{A}$ denotes an
associative unital real algebra. We say a function is
$\mathcal{A}$-differentiable if it is real differentiable and its differential
is in the regular representation of the algebra. We show the theory of
sequences and numerical series resembles the usual theory, but, the proof to
establish this claim requires modification of the standard arguments due to the
submultiplicativity of the norm on $\mathcal{A}$. In contrast, the theorems
concerning divergence of power series over $\mathcal{A}$ are modified notably
from the standard theory. We study how the ratio, root and geometric series
results are modified due to both the submultiplicativity of the norm and the
calculational novelty of zero-divisors. Despite these difficulties, we find
natural generalizations of the usual theorems of real analysis for uniformly
convergent series of functions. We prove the usual calculations with power
series transfer nicely to the $\mathcal{A}$-calculus. Then power series are
used to define sine, cosine, hyperbolic sine, hyperbolic cosine and the
exponential. Many standard identities are derived for entire functions on an
arbitrary commutative $\mathcal{A}$. Finally, special functions are introduced
and we derive the $N$-Pythagorean Theorem which produces $\cos^2 \theta+ \sin^2
\theta=1$ and $\cosh^2 \theta - \sinh^2 \theta = 1$ as well as a host of other
less known identities.
