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? — 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 Power 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/Archaic Form. --prime mover (talk) 10:30, 23 July 2014 (UTC)