Re: Mathematica and infinite series

*To*: mathgroup at smc.vnet.net*Subject*: [mg113200] Re: Mathematica and infinite series*From*: Sam Takoy <sam.takoy at yahoo.com>*Date*: Mon, 18 Oct 2010 05:48:09 -0400 (EDT)

Hi, Thanks for the response! I don't have Mathematica on the computer on which I'm typing this email, so I'm not able to reformulate my question reliably. I was hoping to come up with a series that does not have a closed form analytical expression. I was hoping that Mathematica would be able to treat SeriesCoefficient and an inverse to an infinite Sum. How about (hoping it's "safe" from Mathematica's analytical ability): f[x_] := Sum[BesselJZero[0, n^2] Sin[n^2] Log[Sin[Cos[n]]]Log[n]/(n^2 Factorial[n]) x^n, {n, 1, Infinity}] Assuming[n > 0, SeriesCoefficient[f[x], {x, 0, n}]] Thanks again, Sam ________________________________ From: Andrzej Kozlowski <akoz at mimuw.edu.pl> To: Sam Takoy <sam.takoy at yahoo.com> Sent: Sun, October 17, 2010 12:05:30 PM Subject: [mg113200] Re: [mg113185] Mathematica and infinite series The problem is that Mathematica seems to go crazy when asked to evaluate: Sum[(Log[n]/(n^2*n!))*x^n, {n, 1, Infinity}] E^x*Derivative[1, 0][BellB][-2, x] The answer is given in terms of the derivative of the function BellB[n,x] with respect to the first variable at {-2,x}. But, and this is weird, since Bell[n,x] is the n-th Bell polynomial so n must be a non-negative integer: In[4]:= BellB[-1,2] During evaluation of In[4]:= BellB::intnm: Non-negative machine-size integer expected at position 1 in Subscript[B, -1](2). >> See also here : http://mathworld.wolfram.com/BellPolynomial.html Really weird. Andrzej Kozlowski On 17 Oct 2010, at 12:06, Sam Takoy wrote: > Hi, > > I am about to embark on a project that operates heavily in infinite > series, so I started figuring out Mathematica's basis capabilities. I > found them very impressive, but I came across this: > > > f[x_] := Sum[Log[n]/(n^2 Factorial[n]) x^n, {n, 1, Infinity}] > Assuming[n > 0, SeriesCoefficient[f[x], {x, 0, 4}]] > > > Answer: > > SeriesCoefficient[\!\( > \*UnderoverscriptBox[\(\[Sum]\), \(n = 1\), \(\[Infinity]\)] > \*FractionBox[\( > \*SuperscriptBox[\(x\), \(n\)]\ Log[n]\), \( > \*SuperscriptBox[\(n\), \(2\)]\ \(n!\)\)]\), {x, 0, 4}] > > > Why doesn't Mathematica produce Log[n]/(n^2 Factorial[n]) as the answer? > > Thanks! > > Sam >