STEP 1992 (Paper B)

For 1(ii), surely g(x) can be expanded as a power series with different initial point other than 0 to give "a series of non-negative powers of x"?

Also, xg(x) can't be expanded as a power series with 0 as initial point, so what are they talking/asking about and what am I missing here? Thanks

ok, thanks all for the replies,

I realise that if h(x)=xg(x) then h(0) tends to 1, but how about h'(0), h''(0)? I don't see any easy ways to compute g'(x), g''(x) and so on, I must have missed something. Maple says the expansion is not possible with 0 as initial point, but Maple had given me wrong answers before, so..

I think I'd just do it as power series (for the $\frac{x}{\sinh x}$ bit at any rate).

(i.e. $\frac{\sinh x}{x} = 1 + x^2/6+x^4/120$. so $\frac{x}{\sinh x} = (1+(x^2/6+x^4/120))^{-1}$

$= 1 - (x^2/6+x^4/120) + (x^2/6+x^4/120)^2$

$= 1 - x^2/6 +x^4(1/120+1/36)$).

(I haven't checked the algebra, or even what the power series for sinh x is, but you get the idea...)