The comparison test says:
Let be a series where for all .
(i) If converges and | | for all , then converges
(ii) If and for all , then .
CaptainBlack used the fact that:
and converges by the comparison test, if we take to be and to be
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EDIT 3: I'm so proud of myself ...even though typing that took forever. i guess i'll get faster with practice though, and after i memorize the commands
argument of the form that we actually use if we want to know if this thing
converges, only later will a formal proof be constructed to make the argument rigorous.
The purpose of such an argument is to explain why this converges, which is not always
clear from a formal proof (Sound of Bourbaki turning in his grave from off stage right).
Jhevon has explained what the idea is in more detail