the comparison test. he wasn't being very formal though, since i already answered the question.
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
EDIT: Bare with me, i'm just learning LaTex, it will take me a while to get used to it
EDIT 2: Finally! It is ready!
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
None explicitly, but knowledge of the behaviour of series. This is an informal
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
RonL