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How to compute the series $\displaystyle\sum_{x=0}^\infty\sum_{y=0}^\infty\sum_{z=0}^\infty\frac{1}{2^x(2^{x+y}+2^{x+z}+2^{z+y})}$ ?

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By symmetry, the sum $S$ of this triple series $$ S=\sum_{x,y,z}\frac{1}{2^x\cdot(2^{x+y}+2^{x+z}+2^{z+y})}$$ is also $$ S=\sum_{x,y,z}\frac{1}{2^\color{red}{y}\cdot(2^{x+y}+2^{x+z}+2^{z+y})}=\sum_{x,y,z}\frac{1}{2^\color{red}{z}\cdot(2^{x+y}+2^{x+z}+2^{z+y})}. $$ Furthermore, $$ \frac1{2^x}+\frac1{2^y}+\frac1{2^z}=\frac{2^{x+y}+2^{x+z}+2^{z+y}}{2^{x+y+z}}. $$ Hence, summing these three equivalent formulas for $S$, one gets $$ 3S=\sum_{x,y,z}\frac1{2^{x+y+z}}=\left(\sum_{x}\frac1{2^x}\right)^3, $$ and, finally, $$ S=\frac13\cdot2^3=\frac83. $$ More generally, for every absolutely convergent series $\sum\limits_x\frac1{a_x}$, $$ \sum_{x,y,z}\frac{1}{a_x\cdot(a_xa_y+a_xa_z+a_za_y)}=\frac13\left(\sum_x\frac{1}{a_x}\right)^3. $$

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  • $\begingroup$ That, sir, is very nice. $\endgroup$
    – mez
    Jan 24 '13 at 19:23
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    $\begingroup$ Symmetry is essential here. $\endgroup$
    – Mikasa
    Jan 24 '13 at 19:27
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    $\begingroup$ Awesome! +1 $\;$ $\endgroup$ Jan 24 '13 at 19:40
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    $\begingroup$ Always try to look for symmetry when dealing with double, triple sums ... $\text{nice}^2$ (+1) $\endgroup$ Jan 24 '13 at 21:54
  • $\begingroup$ wow nice answer ... $\endgroup$
    – srijan
    May 15 '13 at 11:33

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