<s>[INST] Find the value of the fraction 
What common fraction is equivalent to $$\frac{\dfrac{1}{1^3}+\dfrac{1}{3^3}+\dfrac{1}{5^3}+\dfrac{1}{7^3}+\cdots}{\dfrac{1}{1^3}+\dfrac{1}{2^3}+\dfrac{1}{3^3}+\dfrac{1}{4^3}+\cdots} \text{ ?}$$

I didn't see how to relate the numerator to the denominator. We can't find the value of the numerator and denominator easily it seems, so what else should we try?
 [/INST] $$\sum_{n=1}^{\infty }\frac{1}{n^3}=\sum_{n=1}^{\infty }\frac{1}{(2n)^3}+\sum_{n=1}^{\infty }\frac{1}{(2n-1)^3}=\zeta(3)$$
so
$$\sum_{n=1}^{\infty }\frac{1}{(2n-1)^3}=\zeta(3)-\frac{1}{8}\zeta(3)=\frac{7}{8}\zeta(3)$$
$$\dfrac{\dfrac{1}{1^3}+\dfrac{1}{3^3}+\dfrac{1}{5^3}+\dfrac{1}{7^3}+\cdots}{\dfrac{1}{1^3}+\dfrac{1}{2^3}+\dfrac{1}{3^3}+\dfrac{1}{4^3}+\cdots}=\frac{\frac{7}{8}\zeta(3)}{\zeta(3)}=???$$
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