Limit Comparison Test
I've been teaching myself Calc. BC this year for the AP test, and this test for infinite series I've proved to be useful in analyzing these things, but I'm not sure how it was derived. I think I'm missing something simple, as the proof in my book is very short, but for this same reason maybe I haven't totally grasped it yet. My Calculus teacher hates infinite series (for whatever reasons) and therefore wasn't much help. Could someone try to break it down for me? Thanks in advance.
If An and Bn are both greater than zero
(lim n->infinity (An/Bn))>0
Then both series converge or diverge.
You should try to find a better teacher!
Suppose that . This means that the sequence can be bounded above and below, say
Multiplying both sides by : . Then we have
If converges, then , so converges by comparison. On the other hand, if converges, then , so converges.
In summary, if the limit condition above holds, then and either both converge or diverge.
Let us know if you have any questions about this.
I'm looking here for the teacher.
Originally Posted by roninpro
I get the "boundness" of the sequence, but going from sequence to series is what gets me.
Why can I say that the sums of the terms will be bounded accordingly too?
I get it, sorry.
If every term is greater or less, than the sum of course will be.
I'm sure you'll see me here again before May.