<p>Improper integrals are really primarily helpful for performing the Integral Test for Series, which is really primarily used for establishing the validity of the p-Series Tests.</p>
<p>Geometric Series and power series are really different entities. A Geometric Series is a series where each term of the series is related by a common ratio, r, where the convergence or divergence of that particular series is determined by the size of |r|. Most geometric series are really comprised of just numerical values, although they don’t have to be.</p>
<p>A power series, on the other hand, is a series of terms of the form a0 + a1(x - c) + a2(x - c)^2 + a3(x - c)^3 + …, where each of the a0, a1, a2, a3, etc. are different constants. A Taylor series is one particular type of power series that represents a function f centered at x = c, where a0 = f(c), a1 = f ‘(c), a2 = f "(c) / 2!, a3 = f "’(c) / 3!, and in general, an = the nth derivative of f (written as f(n)(x) where the (n) is a subscript) evaluated at c. The idea is that the series and the function f share the same concavity, and so over a certain interval, the function and the series converge to that same location, given enough terms.</p>
<p>I can break some of those pieces down further if need be.</p>