Why aren't mathematical series zero-indexed? We're learning about sequences in calculus class, and I keep assuming they are zero-indexed because of my experience in programming. Why aren't they zero-indexed? Can they be zero-indexed?
 A: Even in computer programming, where there are technical reasons to use zero-indexing, the zero-indexing causes confusion. For example, what is the 17th element of a zero-indexed array?  Why, it's a[16], of course! My beginning programming students have often been confused by this mismatch. Off-by-one errors in array indexing are an exceedingly common category of mistake.
The technical reason to prefer zero-indexing in computer applications, already rather slim and unpersuasive, simply does not apply in mathematics.
And  1-indexed sequences are pedagogically and terminologically simpler than 0-indexed sequences.  
In my opinion, the mathematical preference for 1-indexed sequences is clearly justified.
That said, zero-indexed sequences  do often appear in mathematics when they will make the presentation clearer.  For example, consider some sort of machine or process that repeatedly changes from state $s_n$ to state $s_{n+1}$.  It is very common to refer to the initial state of the machine as $s_0$, and then after $i$ transitions the machine is in state $s_i$.
A: A series' indexing set usually starts with the first natural number with which the general term is defined.
If you are asking why in the course of the material, you generally write $\sum_{n\ge1}x_n$ to denote a series, then the reason is that some people define $\mathbb N$ as: $\{1,2,\cdots\}$, others define it as $\{0,1,2,\cdots\}$.
A: Examples of series naturally indexed from $0$ are $$\frac{1}{1-x}=\sum_{k=0}^\infty x^k\;\;(|x|<1)\quad\text{and}\quad\exp x=\sum_{k=0}^\infty\frac{x^k}{k!}\;\;(x\in\Bbb R).$$ Series more naturally indexed from $1$ include $$\ln(1-x)=-\sum_{k=1}^\infty \frac{x^k}{k}\;\;(|x|<1)\quad\text{and}\quad\zeta(x)=\sum_{k=1}^\infty\frac{1}{k^x}\;\;(x>1).$$There is no sustainable argument that one sort of indexation is more fundamental than the other.
