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Alright, I thought this stuff was easy when it was just sequences... like algebra. But now we hit series and stuff and it felt like hitting a brick wall when I tried to do my homework. I couldn't get anywhere. I am like totally lost... Maybe my limit-finding skills are rusty, or maybe I don't understand the new concepts.

Anyhow, here is the very first question in the homework. Directions are to determine whether the series is convergent or divergent and, if CV, to find the sum.

Here's the Series:

I tried taking the limit of it, but l'Hospital's Rule didn't help me much... seems to be indeterminate through and through...

Edit: Oh, btw, I'm only allowed to use the Divergence Test... which can only prove DV and CAN'T prove CV...
which means, how can I be sure that a sequence that fails the DVTest is CV?
>_>

It sounds like a job for *drumroll*: the ratio test!
http://mathworld.wolfram.com/RatioTest.html

I don't know what the divergence test is...

But conceptually, the series would converge if, for high n, the element gets smaller and smaller. Thus, the previous element would have a larger magnitude then the current element. That is the basic premise of the ratio test.

Divergence test

Okay, instead of working straight from [(-3^n-1)/4^n], make it (-1/3) (3/4)^n.
So it becomes Sigma -1/3 * Sigma (3/4)^n.

ah, good eye... so that prooves that it's not divergant, but... how do you find the sum from there?

Edit: Wait, that does prove that it's DV...

Edit2: no wait that's not the limit...

Edit3: Ok, the limit of the sequence is 0, which means that the DV test doesn't help... it could be CV or DV still.

Edit4: Wait... Monotonic Sequence Theorem... the sequence is bounded at 0, so... does that mean that it's CV?

Edit5: Aha, my answer matched the one in the answer key... thanks!

ok, how about this one?


I quickly am able to prove that it is CV (DV test is inconclusive, but the sequence is monotonic [rewrite as ln(n) - ln(2n + 5), so it must be CV), but it's not a geometric series, so I'm unsure where to go from there

This sequence does not converge. Your answer could have been checked in a relatively straightforward manner: Notice that in the limit, the argument of the natural logarithm becomes ~0.5. In other words, if you ignore the first few terms, you are repeatedly summing a non-zero constant (ln 0.5), which will inevitably go to infinity. This means that the series fails the Divergence test since the limit of the individual terms must be equal to zero.