I'm trying to generate a power series expansion of the following function:

 S[x_, l_] := 
   (C[1] + Integrate[E^(2 Sum[t^i/i, {i, 1, l - 1}]) (1 - t)^2 
         Sum[(l - i*2) t^i, {i, 1, l - 1}]/((t - 1) t^l), {t, 1, x}, 
        GenerateConditions -> False])*x^(l - 1)*E^(-2 Sum[x^i/i, {i, 1, l - 1}])/(1 - x)^2;

For l=2 and l=3 it works perfectly, but not in case of l=4:

Table[CoefficientList[Series[S[x, i], {x, 0, 4}], x], {i, 2, 4}]

Is it possible to solve this problem?


The first error says it all. Your S[x,l] does not converge in $x=0$ for $l \ge 3$. Note that evaluating the function in $x=0$ is the first step in computing its series about this point.

If you remove the GenerateConditions -> False option, you will notice that the integration produces some result invalid for $x=0$. By forcing Mathematica to forget this restriction, you can manually insert $x=0$ in the formula but the result can be anything unexpected then. This workaround seems to fix the problem temporarily at $l=3$ but fails at $l=4$.

  • $\begingroup$ In fact S[x,l] is the solution of ODE aimed to soulve recurrence secuence. So, I supposed that computing the series should work for any l. $\endgroup$ – Филипп Цветков Nov 11 '13 at 12:56

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