How to Recognize a Geometric Series What is the "definitive" definition of a geometric series?
I phrased the question this way, because I've checked multiple Calculus textbooks, as well as Paul's Online Math Notes, and they seem to give conflicting definitions. 
Thomas Calculus (12th ed) says, "Geometric Series are series of the form:"
$$\sum_{n=1}^{\infty}ar^{n-1}$$
Calculus for Scientists and Engineers says, "Geometric Series have the form:"
$$\sum_{k}^{\infty} r^k \space \text{or} \space \sum_{k}^{\infty}ar^k$$
and that $a$ is simply the first term.
My textbook, Stewart Calculus Early Transcendentals (7th ed) says, "An important example of an infinite series is the geometric series"
$$a+ar+ar^2+...+ar^{n-1}+... = \sum_{n=1}^{\infty} ar^{n-1}$$
But then Stewart goes on to provide examples and exercises which do not fit that form, such as:
$$\sum_{n=1}^{\infty}ar^n$$
Paul's Online Math Notes say that a geometric series must have one of the forms: 
$$\sum_{n=0}^{\infty} ar^n \space \text{or} \space \sum_{n=1}^{\infty} ar^{n-1}$$
This was all especially confusing for me while trying to do homework, and in class, when we were shown the following problem:
$$\text{Determine whether the following series converges or diverges, and if it converges, find the sum:} \space \sum_{n=1}^{\infty} \frac{1}{2^n}$$
I recognized the common ratio $r=\frac{1}{2}$, but I thought that $a=1$. It turned out, however, that $a=\frac{1}{2}$, also.
This would have made sense to me if the series had had the form $\sum_{n=1}^{\infty}ar^{n-1}$, but it didn't.
In other words, if the index begins with $n=1$, how can you end up with a series of terms $a+ar+ar^2+ar^3+...$? The first term would be $ar^1=ar$; there's no way you could have $ar^0=a(1)$, when the index begins with 1.
And if that's the case, how can $\frac{1}{2}$ be simultaneously equal to $a$ and to $r$, when there is only a single term in the series definition itself? 
Is there a definitive, universal definition for a Geometric Series, or is it somewhat subjective and/or open to interpretation? Or am I simply missing something here?
 A: The definition of a geometric series is a series where the ratio of consecutive terms is constant.  It doesn't matter how it's indexed or what the first term is or whether you have a constant.  That stuff just has to do with how you write the series.  So the '$a$' in a given series need not be unique, as you pointed out.  But the common ratio $r$ is uniquely determined by the series. $$\sum\limits_{n=0}^{\infty} a r^n$$ $$\sum\limits_{n=3}^{\infty} r^n$$ $$\frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \cdots$$ $$1+1+1+ \cdots$$ are all examples of geometric series (the last one doesn't converge).  For example, the first one is a geometric series because the ratio of consecutive terms is $$\frac{ar^{k+1}}{ar^k} = r$$
A: All the definitions here are equivalent :
$$\sum_{i=1}^{\infty}{ar^{i-1}}=ar^0+ar^1+ar^2+...=\sum_{j=0}^{\infty}{ar^j} ~~~~\text{where}~(j = i-1)$$
Moreover if $a = r$, then $ar^0 = a = r = r^1$ so
$$\sum_{p=0}^{\infty}{ar^p}=\sum_{q=1}^{\infty}{r^q} ~~~~~\text{where}~(p=q-1)$$
But this is indeed confusing when we are used to finite sum,
where, for $M\in \mathbb{N^*}$
$$\sum_{k=1}^{M}{ar^{k-1}} = ar^0+ar^1+...+ar^{M-1}=\sum_{m=0}^{M-1}{ar^m} ~~~~~\text{where}~(m = k-1)$$
In your instance
$$\frac{1}{2^n}=\left(\frac{1}{2}\right)^n$$
So you could choose $r_1$ to be $\frac{1}{2}$ and write your serie as
$$(a_1,r_1)\leftarrow(1,\frac12) ~~~~~\sum_{i=1}^\infty{r_1^i}$$
or choose $a_2$ to be $\frac12$ and write
$$(a_2,r_2)\leftarrow (\frac12,\frac12) ~~~~~ \sum_{i=0}^{\infty}{a_2r_2^i}$$
A: Shift the index and subtract
$$ \sum_{n=1}^\infty \frac{1}{2^n} = \sum_{n=0}^\infty \frac{1}{2^n} - 1
$$
or, you can multiply and multiply
$$ \sum_{n=1}^\infty \frac{1}{2^n} = \frac{1}{2}\sum_{n=0}^\infty \frac{1}{2^n}
$$
