Re: Series expansion of Lambert series

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• Subject: [mg128011] Re: Series expansion of Lambert series
• From: "danl at wolfram.com" <daniel.lichtblau0 at gmail.com>
• Date: Sat, 8 Sep 2012 03:11:28 -0400 (EDT)
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```On Friday, September 7, 2012 3:56:27 AM UTC-5, Dr. Wolfgang Hintze wrote:
> Define for 0<=x<1 the Lambert series
>
>
>
> f[x_] := Sum[ x^n/(1-x^n), {x,1,oo}]
>
>
>
> 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}]
>
>
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> does not seem to be expandable with Series.
>
>
>
> BTW same thing in 8.0 and 5.2
>
>
>
> Regards,
>
> Wolfgang

Use the fact that you only need n terms of the sum (in these specific examples) in order to obtain n terms of the series expansion.

In[17]:= seriesLambert[x_,n_] := Series[Sum[x^k/(1-x^k),{k,n}], {x,0,n}]

In[18]:= seriesLambert[x,4]

2      3      4       5
Out[18]= x + 2 x  + 2 x  + 3 x  + O[x]

Daniel Lichtblau
Wolfram Research

```

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