Geometric series
Initial term a

The geometric series a + a r + a r 2 + a r 3 + … {\displaystyle a+ar+ar^{2}+ar^{3}+\dots } has the same coefficient a {\displaystyle a} in every term. The first term of a geometric series is equal to this coefficient and is the parameter a {\displaystyle a} of that geometric series, giving a {\displaystyle a} its common interpretation: the "initial term."

This initial term defines the units of measurement of the series as a whole, if it has any, and in applications it will often be named according to a noun with those units. For instance a {\displaystyle a} could be an "initial mass" in a radioactive decay problem, with units of mass of an isotope, an "initial payment" in mathematical finance, with units of some type of currency, or an "initial population" in demography or ecology, with units of a type such as nationality or species.

In generator form, ∑ k = 0 ∞ a r k, {\displaystyle \sum _{k=0}^{\infty }ar^{k},} this term is technically written a r 0 {\displaystyle ar^{0}} instead of the bare a {\displaystyle a}. This is equivalent because r 0 = 1 {\displaystyle r^{0}=1} for any number r. {\displaystyle r.}

In contrast, a general power series a 0 + a 1 r + a 2 r 2 + a 3 r 3 + … {\displaystyle a_{0}+a_{1}r+a_{2}r^{2}+a_{3}r^{3}+\dots } would have coefficients a k {\displaystyle a_{k}} that could vary from term to term. In other words, the geometric series is a special case of the power series. Connections between power series and geometric series are discussed below in the section § Connections to power series.
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