Re: Can Mathematica do this (infinite series)?

*To*: mathgroup at smc.vnet.net*Subject*: [mg113421] Re: Can Mathematica do this (infinite series)?*From*: Leonid Shifrin <lshifr at gmail.com>*Date*: Thu, 28 Oct 2010 04:25:55 -0400 (EDT)

Hi Sam, One way would be to use local rules: ClearAll[f, g]; g[x_] := f[x]^2 + Sin[x] f[x] dcrules = {f[0] :> c[0], Derivative[n_][f][0] :> c[n]*n!, c[n_] :> Log[n + 1/3]/n!}; Normal[Series[g[x], {x, 0, 5}] //. dcrules] Log[3]^2 + x (-Log[3] - 2 Log[4/3] Log[3]) + x^2 (Log[4/3] + Log[4/3]^2 - Log[7/3] Log[3]) + 1/6 x^3 (3 Log[7/3] + 6 Log[4/3] Log[7/3] + Log[3] - 2 Log[3] Log[10/3]) + 1/12 x^4 (-2 Log[4/3] + 3 Log[7/3]^2 + 2 Log[10/3] + 4 Log[4/3] Log[10/3] - Log[3] Log[13/3]) + 1/120 x^5 (-10 Log[7/3] - Log[3] + 20 Log[7/3] Log[10/3] + 5 Log[13/3] + 10 Log[4/3] Log[13/3] - 2 Log[3] Log[16/3]) In[16]:= % // N Out[16]= 1.20695 - 1.73071 x - 0.560409 x^2 + 0.409604 x^3 + 0.313403 x^4 + 0.155857 x^5 You can do SeriesCoefficient as well. Regards, Leonid On Tue, Oct 26, 2010 at 1:34 PM, Sam Takoy <sam.takoy at yahoo.com> wrote: > Hi, > > I'm working on a project involving infinite series and I don't know how > to do it or even ask a sensible question about it. So I cooked up a > question the answer to which might give me ideas. > > f[x_]:=Sum[c[n]x^n, {n, 1, Infinity}] > > What's the infinite series for > > f[x]^2 + Sin[x]f[x] > > in terms of c[n]? > > What's the simplest way that Mathematica can answer this question for > general c[n]? > > The pipe dream is this: > > f[c_][x_] := Sum[c[n] x^n, {n, 1, Infinity}] > > g[c_][x_] := f[c][x]^2 + Sin[x] f[c][x] > > d[c_][n_] := SeriesCoefficient[g[c][x], {x, 0, n}] > > d[Log[# + 1/3]/Factorial[#] &][5] // N > > Thanks, > > Pavel > >