Question

If $\underset{( i \neq j \neq k )}{\displaystyle\sum_{i=0}^{\infty} \displaystyle\sum_{j=0}^{\infty} \displaystyle\sum_{k=0}^{\infty} \frac{1}{3^i 3^j 3^k}}=\frac{27 \times A}{208}$; where $A$ is single digit natural number, then find the value of $A$

Answer 1

No condition on $i , j , k$
$=\frac{3}{2} \times \frac{3}{2} \times \frac{3}{2}=\frac{27}{8}$
when any two of $i, j, k$ are equal
$=\displaystyle\sum_{ i =0}^{\infty} \displaystyle\sum_{ j =0}^{\infty} \frac{1}{3^{2 i }} \cdot \frac{1}{3^{ j }}=\frac{3}{2}\left[1+\frac{1}{9}+\ldots \ldots .\right]=\frac{3}{2} \times \frac{9}{8}=\frac{27}{16}$
When all three are equal
$=1+\frac{1}{27}+\frac{1}{27^2}+\ldots \ldots \infty $
$ =\frac{27}{26} $
$\text { Required } =\frac{27}{8}-3 \cdot \frac{27}{16}+\frac{2 \cdot 27}{26}=\frac{81}{208}$