Question : for what values of a does the following series converge ?\[S=1+3a+a^2+3a^3+a^4+3a^5+a^6+\ldots\]Solution :Let Given series as \[S=1+3a+a^2+3a^3+a^4+3a^5+a^6+\ldots\] This series can be factored as \[S\ =\ (1+a^2+a^4+\ldots)+3a\left(1+a^2+a^4+..\right)\] \[S=\left(1+3a\right)\left(1+a^2+a^4+\ldots\right)\] Since 1+3a is finite for all b and \[1+a^2+a^4+\ldots\] is a geometric…
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