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Re: Mathematica and infinite series

  • To: mathgroup at smc.vnet.net
  • Subject: [mg113201] Re: Mathematica and infinite series
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Mon, 18 Oct 2010 05:48:20 -0400 (EDT)

No, this sort of thing certainly won't work. Unless Mathematica can find a 
closed form of the sum it will not be able to compute a general series coefficient from an infinite series. However, it will work if you truncate the series than you can get particular series coefficients, like this:

f == Sum[BesselJZero[0, n^2] Sin[n^2] Log[
      Sin[Cos[n]]] Log[n]/(n^2 Factorial[n]) x^n, {n, 1, 10}] +
   O[x]^11;

SeriesCoefficient[f, 4]

and so on.

Andrzej

On 17 Oct 2010, at 21:44, Sam Takoy wrote:

> 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: 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
> >
>
>
>


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