Difficult: Awesome, so I understood why we use sequences of partial sums, but will we always need to prove two lemmas to prove convergence? The first lemma to prove {s_n} can be expressed in terms of n and the second to prove that the limit is convergent to a limit? I guess once we have those, we just plug those limits in and the proof is pretty easy.
And, holy cow, it looks tricky to prove a sequence diverges! Maybe it was just because the harmonic sequence is specifically difficult to prove, But those algebraic manipulations didn't seem intuitive at all. Is there a general structure for a proof that a series is divergent? Or does it always just depend on what the series is?
Interesting: Well, I've always used the words series and sequences interchangeably, but now I know the difference! A series is just the numbers in a row, where a series adds them all together! I've also hear of the harmonic sequence before, so it's really cool to finally know what it is. I can't believe it diverges, it seems like it wouldn't! Oh well, my intuition has definitely deceived me before.
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