Re: Series expansion of Lambert series

*To*: mathgroup at smc.vnet.net*Subject*: [mg128014] Re: Series expansion of Lambert series*From*: michael partensky <partensky at gmail.com>*Date*: Sat, 8 Sep 2012 03:12:28 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net*References*: <20120907085449.BF69F6790@smc.vnet.net>

Hi, Wolfgang. The first definition seems ambiguous. (1) x in (1) is a "silent" variable (summation index), and can not appear in the left side. (2) x=1 leads to divergence (~\inf). And it's contradict the allowed range of x. (3) Apparently, the summation index is n, in which case the next comment can be helpful. As to Eq. (2), try this: e1[x_] := NSum[x^n/(1 + Factorial[n]), {n, 0, \[Infinity]}]; In[12]:= Table[e1[x], {x, 1, 10}] Out[12]= {1.52607, 5.00291, 15.7894, 47.4019, 136.98, 385.988, \ 1070.88, 2943.92, 8050.98, 21954.5} Hope it helps. Best. Michael Partenskii olfgang Hintze <weh at snafu.de> wrote: > Define for 0<=x<1 the Lambert series > > f[x_] := Sum[ x^n/(1-x^n), {x,1,oo}] *(1)* > > How do I get Mathematica to calculate the first few terms of a series > expansion like > > Series[f[x],{x,0,4}] ? > > Also a series like > > e1[x_] := Sum[x^n/(1+n!), {n,0,oo}] * (2)* > > does not seem to be expandable with Series. > > BTW same thing in 8.0 and 5.2 > > Regards, > Wolfgang > >

**References**:**Series expansion of Lambert series***From:*"Dr. Wolfgang Hintze" <weh@snafu.de>