<s>[INST] $\frac{1+\frac{1}{2^{11}}+\frac{1}{3^{11}}+\frac{1}{4^{11}}+...}{1-\frac{1}{2^{11}}+\frac{1}{3^{11}}-\frac{1}{4^{11}}+...}$ I'm not sure how to solve this summation and would like assistance.  I believe the answer is ||$\frac{1024}{1023}$|| according to Wolfram.  I don't know what the proper approach is (creating two summations and somehow combining or common denominators..?).
 [/INST] I will assume that the +1 in the denominator is a -1 and go on,
we can write $\frac{1}{1}+\frac{1}{2^{11}}+\frac{1}{3^{11}}+.......=S$
now we can take the even terms to one side and express it as
$\frac{1}{1}$+$\frac{1}{3^{11}}$+$\frac{1}{5^{11}}$=S +$\frac{1}{2^{11}}(S)$
$\frac{1}{1}+\frac{1}{3^{11}}+\frac{1}{5^{11}}=S(\frac{2^{11}-1}{2^{11}})$
now the equation in the question becomes 
$\frac{S}{S-2(S(\frac{2^{11}-1}{2^{11}})}$ which gives the answer $\frac  {1024}{1023}$
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