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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?

Jared Silver
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    Short answer: because the author of your book didn't want to, and yes they can. – Arthur Mar 23 '15 at 18:31
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    Someone a long time ago thought that when you count, you begin with $1$. Strange but true. – GEdgar Mar 23 '15 at 18:32
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    Along with the above reasons, I think it might be the case that simplicity is also a factor: usually we have terms like $\frac{1}{n}$ or $\ln(n)$, etc. For $n=0$ these are undefined, so to avoid writing $n+1$ everywhere, we instead just start at $n=1$. Of course, you could replace $n$ with $n+1$ everywhere and start at $0$, or alternatively just make the $0$-th term $0$. – Hayden Mar 23 '15 at 18:36
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    In programming they are indexed with zero because it represents an offset in memory. In general mathematical usage you can index a sequence with anything, zoo animals if you like. – DanielV Mar 23 '15 at 18:36
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    some languages (e.g., MATLAB and R) are 1-based too – MichaelChirico Mar 23 '15 at 18:37
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    $a_{\text{aardvark}}=1, a_{\text{baboon}}=2, a_{z+1}=a_z+a_{z-1}$. What is $a_{\text{rabbit}}$? – abiessu Mar 23 '15 at 18:39
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    http://en.wikipedia.org/wiki/On_Beyond_Zebra! – Will Jagy Mar 23 '15 at 18:41
  • Why not use $\mathbb{Z}$ indexing instead? – copper.hat Mar 23 '15 at 19:01
  • Sometimes it does make sense to use $\Bbb Z$ indexing. For example, if $f_i$ is the $i$th fibonacci number, with $f_0 = 0$ and $f_1=1$, then there is a unique extension of $f$ to the integers that conserves the property $f_i + f_{i+1} = f_{i+2}$ for all $i$. This is a natural thing to do, because identities like $f_mf_{n+1} + f_{m-1}f_n = f_{m+n}$ still hold even when $m$ or $n$ is negative. – MJD Jul 01 '23 at 19:05

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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.

John Bentin
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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 $s_i$.

MJD
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  • Maybe f77 was ahead of its time :-). – copper.hat Mar 23 '15 at 19:00
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    It was so far ahead of its time, it invented the 1-based arrays in 1955! – MJD Mar 23 '15 at 19:02
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    I’m not aware of any mathematical preference for $1$-based indexing. Ceteris paribus I prefer $0$-based indexing, though that may be my background in set theory showing. Whether $1$-based sequences are pædagogically simpler depends very much on the sequence and what’s being done with it. – Brian M. Scott Mar 24 '15 at 01:27
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    @MJD, linguistically, I think the key is just to avoid the phrase "17th element" and just say "element 17", meaning $a(17)$. The first element is element $0$, of course. – goblin GONE Aug 16 '16 at 12:06
  • You have to climb 16 storeys of a building to get to the 17th floor. The 17th floor is at Storey[16]. – Kaz Jun 29 '23 at 05:21
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    Rulers have to be zero based. You cannot have the first hatch of a ruler be simultaneously 1 mm and 1 cm. Computer memory calculations use rulers, because multiple units are used simultaneously. If the first word is 1, and the first byte is 1, you cannot convert words to bytes just by multiplying. Just like you can't convert cm to mm by multiplying if the first hatch of the ruler isn't zero. – Kaz Jun 29 '23 at 05:23
  • I take your point, and I will stand by my statement “The technical reason to prefer zero-indexing in computer applications…, simply does not apply in mathematics.” – MJD Jun 29 '23 at 13:48
  • zero based indices allow you to use $\sum_{i \in n}$ instead of $\sum_{i = 1}^{n}$ since $n = { m \in \mathbb{N} | m < n }$ – Poscat Oct 19 '23 at 02:58
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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 $\{0,1,2,\cdots\}$.