Question

If $\displaystyle\sum_{ i =0}^{\infty} \displaystyle\sum_{ j =0}^{\infty} \frac{1}{ a ^{ i } \cdot a ^{ j }}=\frac{\lambda a ^2}{( a -1)^2( a +1)}$ where $i \neq j$ and $a >1$ then possible values of $\lambda$ may be

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (C), (D)

When no restriction on i and j
$S =\displaystyle\sum_{ i =0}^{\infty} \displaystyle\sum_{ j =0}^{\infty} \frac{1}{ a ^{ i } \cdot a ^{ j }}=\left(1+\frac{1}{ a }+\frac{1}{ a ^2}+\ldots \infty\right)^2=\frac{ a ^2}{( a -1)^2} i \neq j$
$\Rightarrow S =\frac{\lambda a ^2}{( a -1)^2( a +1)} \Rightarrow \lambda>2$