Maclaurin series and general terms

Question

For a general term in a maclaurin (or any other) series e.g. x^r/r! is it always that r=0, or r=1 at the first term or does it have nothing to do with the term it appears in e.g. r=10 could come before the term when r=9, I ask because my formula book changes from r=0 for the first to r=1?

Answer 1

It depends on the function. The r=0 term may have a zero coefficient, in which case, you can skip it. Some functions may have many zero terms before the series actually starts.

The order of terms is usually given in increasing powers of $x$. You may exchange some terms at the beginning of the series, but how would you write it in a practical $\sum$ notation? And why would you mess with the order?

Note that while permuting some early terms (like exchanging r=9 and r=10) has no effect, you shouldn't reorder them in a way that messes with the order all the way to infinity - this can make the series divergent, or worse, make it converge to the wrong value (conditional convergence).