Question

${\sum }_{i=1}^{\infty }{\sum }_{j=1}^{\infty }{\sum }_{k=1}^{\infty }\frac{1}{a^{i+j+k}}$ is equal to (where $a>1$) :

• A. $(a-1{)}^{-3}$
• B. $\frac{1}{a^3-1}$
• C. $\frac{3}{a-1}$
• D. None of the above

Answer 1

The correct option is A $(a-1{)}^{-3}$

We know the value of the infinite summation ${\sum }_{k=0}^{\infty }\frac{1}{a^k}=\frac{1}{a-1}$,

Now,

${\sum }_{i=1}^{\infty }{\sum }_{j=1}^{\infty }{\sum }_{k=1}^{\infty }\frac{1}{a^{i+j+k}}⟹{\sum }_{i=1}^{\infty }{\sum }_{j=1}^{\infty }\frac{1}{a^{i+j}}\frac{1}{a-1}⟹{\sum }_{i=1}^{\infty }\frac{1}{a^i}\frac{1}{{(a-1)}^2}⟹\frac{1}{{(a-1)}^3}$