Talk:Primitive of x over Sine of a x

What's $n$? --GFauxPas (talk) 00:01, 14 July 2014 (UTC)


 * The $n$th element, of course, didn't think it needed to be specified, it's kind of obvious. --prime mover (talk) 07:47, 14 July 2014 (UTC)


 * Let me put it this way. In $\displaystyle \left({a x + \frac {\left({a x}\right)^3} {18} + \frac {7 \left({a x}\right)^5} {1800} + \cdots + \frac {2 \left({2^{2 n - 1} - 1}\right) B_n \left({a x}\right)^{2 n + 1} } {\left({2 n + 1}\right)!} }\right)$, how do you know how many terms to use? --GFauxPas (talk) 11:37, 14 July 2014 (UTC)


 * Yes, what is $n$ in terms of the LHS? &mdash; Lord_Farin (talk) 16:45, 14 July 2014 (UTC)


 * Haven't a clue. It would become apparent at the point at which the proof is demonstrated. As it is, this is how it appears in the source work. --prime mover (talk) 22:01, 14 July 2014 (UTC)


 * ... on the other hand, the index goes $1, 3, 5, \ldots, 2n + 1, \ldots$ so it is sort of easily inferred. --prime mover (talk) 22:03, 14 July 2014 (UTC)

O sorry, yes I get you now, I forgot the $+ \cdots$ --prime mover (talk) 22:04, 14 July 2014 (UTC)


 * While we're at it. Should that sum not be alternating in sign? And should that not be the $2n$th Bernoulli number, at least by the cited definition? --Ybab321 (talk) 07:23, 15 July 2014 (UTC)


 * This is how it appears in the cited work. I haven't worked through it (because laziness) but both your points may be valid.  The source work has been known to print errors.  The suggested method of solution looks to me like: do a taylor expansion of cosecant, reduce the index of x in each term by one, then integrate termwise.


 * Looking at Taylor Series Expansion for Tangent Function, one supposes the cosecant expansion may follow similar lines, but without ploughing through it I'm not in a position to say. --prime mover (talk) 11:16, 15 July 2014 (UTC)


 * This look reasonable? --Ybab321 (talk) 14:29, 17 July 2014 (UTC)


 * Looks like so. As I say, Spiegel's been egregiously wrong before, so probably is here too. We need a page for Power Series Expansion for Cosecant Function, and then we'll have that rock solid. I will also check the definition of Bernoulli numbers as reported in this manual, see whether there's a different definition being used. --prime mover (talk) 16:09, 17 July 2014 (UTC)

This question has resolved itself by consulting Spiegel and noting his definition, documented here: Definition:Bernoulli Numbers/Also defined as. --prime mover (talk) 10:30, 23 July 2014 (UTC)