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In this paper we study sequences, series, power series and uniformconvergence in the $\mathcal{A}$-Calculus. Here $\mathcal{A}$ denotes anassociative unital real algebra. We say a function is$\mathcal{A}$-differentiable if it is real differentiable and its differentialis in the regular representation of the algebra. We show the theory ofsequences and numerical series resembles the usual theory, but, the proof toestablish this claim requires modification of the standard arguments due to thesubmultiplicativity of the norm on $\mathcal{A}$. In contrast, the theoremsconcerning divergence of power series over $\mathcal{A}$ are modified notablyfrom the standard theory. We study how the ratio, root and geometric seriesresults are modified due to both the submultiplicativity of the norm and thecalculational novelty of zero-divisors. Despite these difficulties, we findnatural generalizations of the usual theorems of real analysis for uniformlyconvergent series of functions. We prove the usual calculations with powerseries transfer nicely to the $\mathcal{A}$-calculus. Then power series areused to define sine, cosine, hyperbolic sine, hyperbolic cosine and theexponential. Many standard identities are derived for entire functions on anarbitrary commutative $\mathcal{A}$. Finally, special functions are introducedand 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 otherless known identities.

Theory of Series in the $\mathcal{A}$-calculus and the $N$-Pythagorean Theorem