Calculation of the vacuum polarization, $<phi^2(x)>$, and expectation value of the stress tensor, $<T_{mu u}(x)>$, has seen a recent resurgence, notably for black hole spacetimes. To date, most calculations of this type have been done only in four dimensions. Extending these calculations to $d$ dimensions includes $d$-dimensional renormalization. Typically, the renormalizing terms are found from Christensens covariant point splitting method for the DeWitt-Schwinger expansion. However, some manipulation is required to put the correct terms into a form that is compatible with problems of the vacuum polarization type. Here, after a review of the current state of affairs for $<phi^2(x)>$ and $<T_{mu u}(x)>$ calculations and a thorough introduction to the method of calculating $<phi^2(x)>$, a compact expression for the DeWitt-Schwinger renormalization terms suitable for use in even-dimensional spacetimes is derived. This formula should be useful for calculations of $<phi^2(x)>$ and $<T_{mu u}(x)>$ in even dimensions, and the renormalization terms are shown explicitly for four and six dimensions. Furthermore, use of the finite terms of the DeWitt-Schwinger expansion as an approximation to $<phi^2(x)>$ for certain spacetimes is discussed, with application to four and five dimensions.
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