<s>[INST] Compute $\sum\limits_n(-1)^{n-1}\frac{2^{n+1}}{2^{2n}-1}$ in terms of $\sum\limits_n\frac1{2^n-1}$ and $\sum\limits_n\frac1{2^n+1}$ Question: 

let  $$\sum_{n=1}^{+\infty}\dfrac{1}{2^n-1}=E,\sum_{n=1}^{+\infty}\dfrac{1}{2^n+1}=F$$ where $E,F$ are constant,(in fact,$E$ is  Erdős-Borwein Constant ),Find the sum
  $$f=\sum_{n=1}^{+\infty}(-1)^{n-1}\dfrac{2^{n+1}}{2^{2n}-1}$$

I tried this which I found in my texbook which seems relative, but Im not sure how to apply it to the problem:
$$f=\sum_{n=1}^{+\infty}(-1)^{n-1}\left(\dfrac{1}{2^n+1}+\dfrac{1}{2^n-1}\right)=\sum_{n=1}^{+\infty}\dfrac{(-1)^{n-1}}{2^n-1}+\sum_{n=1}^{+\infty}\dfrac{(-1)^{n-1}}{2^n+1}$$ following can't try
 [/INST] Let $d(m),d_1(m),d_0(m)$ be the number of divisors, odd divisors, even divisors of $m$. We have:
$$ F=\sum_{n\geq 1}\left(\frac{1}{2^n}-\frac{1}{2^{2n}}+\frac{1}{2^{3n}}-\ldots\right)=\sum_{m\geq 1}\frac{d_1(m)-d_0(m)}{2^m}=2\sum_{m\geq 1}\frac{d_1(m)}{2^m}-E $$
$$ E = \sum_{m\geq 1}\frac{d(m)}{2^m} $$
On the other hand:
$$\begin{eqnarray*}f=\sum_{n\geq 1}(-1)^{n+1}\frac{2^{n+1}}{(2^n-1)(2^n+1)}&=&\sum_{n\geq 1}(-1)^{n+1}\left(\frac{1}{2^n-1}+\frac{1}{2^n+1}\right)\\&=&2\sum_{n\geq 1}(-1)^{n+1}\left(\frac{1}{2^n}+\frac{1}{2^{3n}}+\frac{1}{2^{5n}}+\ldots\right)\\&=&2\sum_{m\geq 1}\frac{1}{2^m}\sum_{\substack{d\mid m \\ d\text{ odd}}}(-1)^{\frac{m}{d}+1}\\&=&2\sum_{\substack{m\geq 1\\m\text{ odd}}}\frac{d(m)}{2^m}-2\sum_{\substack{m\geq 1\\m\text{ even}}}\frac{d_1(m)}{2^m}\\&=&4\sum_{\substack{m\geq 1\\m\text{ odd}}}\frac{d(m)}{2^m}-2\sum_{m\geq 1}\frac{d_1(m)}{2^{m}}\end{eqnarray*} $$
and
$$ E+F=2\sum_{m\geq 1}\frac{d_1(m)}{2^m}=2\sum_{\substack{m\geq 1\\m\text{ odd}}}d(m)\left(\frac{1}{2^m}+\frac{1}{2^{2m}}+\frac{1}{2^{4m}}+\frac{1}{2^{8m}}+\ldots\right) $$
so 
$$ f=\sum_{\substack{m\geq 1 \\ m\text{ odd}}}d(m)\left(\frac{2}{2^m}-\frac{2}{2^{2m}}-\frac{2}{2^{4m}}-\frac{2}{2^{8m}}-\frac{2}{2^{16m}}-\ldots\right) $$
$$ E=\sum_{\substack{m\geq 1 \\ m\text{ odd}}}d(m)\left(\frac{1}{2^m}+\frac{2}{2^{2m}}+\frac{3}{2^{4m}}+\frac{4}{2^{8m}}+\frac{5}{2^{16m}}+\ldots\right) $$
$$ F=\sum_{\substack{m\geq 1 \\ m\text{ odd}}}d(m)\left(\frac{2}{2^m}+\frac{0}{2^{2m}}-\frac{1}{2^{4m}}-\frac{2}{2^{8m}}-\frac{3}{2^{16m}}+\ldots\right) $$
and I do not see any way to write $f$ in terms of $E$ and $F$ only, without involving $\sum_{\substack{m\geq 1\\m\text{ odd}}}\frac{d(m)}{2^m}$ too.
In any case the numerical evaluation of such (Lambert) series is pretty simple: an acceleration technique is outlined here.
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