For the 3^i-1 and 3^1+2 it's 3 to the exponent i-1 and i+2 respectively not just i.
(e^i)/(3^(i-1)) = 3(e/3)^i. And since (e/3)<1 , lim(i-->inf.)3(e/3)^i = 0.
2^iπ^i/((3^(i+2)) = (1/9)(2π/3)^i and because, (2π/3) >1 , lim(i-->inf)(1/9)(2π/3)^i = infinity.
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(e^i)/(3^(i-1)) = 3(e/3)^i. And since (e/3)<1 , lim(i-->inf.)3(e/3)^i = 0.
2^iπ^i/((3^(i+2)) = (1/9)(2π/3)^i and because, (2π/3) >1 , lim(i-->inf)(1/9)(2π/3)^i = infinity.