What are the advantages of using one over the other? I mean this in the context of sequences and series. For example, should we let the geometric sequence start from $n=0$ or $n=1$ to get $a_n = a_0r^n$ or $a_n = a_1r^{n-1}$, respectively? Another example is the arithmetic sequence, which changes from $a_0 + dn$ to $a_1 + d(n-1)$ based on which number we start with (0 or 1).
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The only advantage of the former, as far as I know, is shorter expressions – Yuriy S Jul 10 '16 at 14:17
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1In Taylor series $\sum_0^\infty b_n(x-a)^n$ is the natural way to proceed. – André Nicolas Jul 10 '16 at 14:19
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Sometimes one convention is simpler and more natural, and sometimes the other. It is probably best to stick with the convention that you are presented with, if that is a normal one, unless there is a good reason to alter it. It is a good idea not to chop and change from one to the other in a similar context (e.g. arithmetic or geometric sequences): otherwise you may end up forgetting which convention was assumed by the formula that you remember.
Edit: See this answer to a related question for specific examples.
John Bentin
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Most computer scientists might argue that numbering should start at zero.
parsiad
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A computer scientist told me long ago: “Computers count from zero; people count from one.” – amd Jul 10 '16 at 20:44
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In most programming languages designed for mathematics, indexing begins at 1. Matlab, R, Mathematica, and Fortran are all good examples. – Underminer Feb 14 '18 at 16:58