A) \[\frac{a}{1-a}+\frac{ab}{1-ab}\]
B) \[\frac{{{a}^{2}}}{1-{{a}^{2}}}+\frac{ab}{1-ab}\]
C) \[\frac{b}{1-b}+\frac{a}{1-a}\]
D) \[\frac{{{b}^{2}}}{1-{{b}^{2}}}+\frac{ab}{1-ab}\]
Correct Answer: B
Solution :
We have, \[|a|<1,|b|<1\] \[\therefore \] \[|ab|=|a|b|<1\] Now \[a(a+b)+{{a}^{2}}({{a}^{2}}+{{b}^{2}})+{{a}^{3}}({{a}^{3}}+{{b}^{3}})+...\] \[=[({{a}^{2}}+{{a}^{4}}+{{a}^{6}}+...)]+[\{ab+(a{{b}^{2}})+(a{{b}^{3}})+...\}]\] \[=\frac{{{a}^{2}}}{1-{{a}^{2}}}+\frac{ab}{1-ab}\]
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