Do any of you know a good technique for partial fraction expansion? I'd love to know of a source that explains the reasoning behind it as the methods I've seen in Stewart's Calculus (an appendix) are very mechanical and I have trouble memorizing by rote.
I can't recall the techniques I've seen before (that's the problem). I'm hoping to really grok how it works so I can derive the solution instead of memorizing
@WilliamGrobman Denominator should have distinct roots. That's the only one I can remember. // Also it works only if you have a bunch of linear factors. For quadratics, it does not work.
@HenningMakholm Isn't that question like equivalent to Cantor intersection theorem? If the OP doesn't know it, should we have to reinvent the wheel and prove the latter theorem?
@Srivatsan I'm not completely sure what the canonical statement of the intersection theorem is -- I think it might be about a strictly decreasing (by inclusion) sequence of closed intervals. That is, of course, easily equivalent to the OP's version, but the equivalence is not completely vacuous.
@HenningMakholm Not vacuous, but not too difficult. If we do not state the intersection theorem for general compact sets and restrict ourselves to intervals, then the OP's question is only more general. The other direction is a bit more involved, but if we define C_N to be the intersection of [a_n, b_n] for n <= N, then we can apply regular CIT and conclude the OP's statement.
@Srivatsan That much is obvious to you and me, but perhaps not to the OP. Anyway the question is academic, as the OP disclaims knowledge of any kind of CIT. Let's see if the comment hint I've provided will clear it up.
@Martin: When you want to add tag wiki/excerpt it is not just "for the sake of adding them", the wiki itself should contain more than a single line that you have put on the excerpt.
Although, maybe it's a bit misleading to say that category theory studies structures by properties of objects and morphisms. That's like saying group theory studies group elements by their multiplication table.
Well, when a student first encounters category theory, the most important difference from what he saw before is that everything is defined using objects, morphisms, arrows, universal properties....
It occurs to me that I don't really know how to explain what category theory is about. It's easy enough to explain what a category is and how to interpret it, and it's easy to say that category theory is the study of categories... but it doesn't really give the motivations for doing so.
I asked "What is confusing" and he said "Mainly how you're using this sum notation for a proof by induction. But I seem to have gotten through it without this notation. Like I had mentioned, it's for a homework assignment, so I'm still learning discrete math. I don't get much practice with it either"
I guess it's just a case of "I've already solved it, I'm not going to dedicate time to thinking more about it"
@Matt Er, I am completely ignorant about dogs. Plus -- to repeat myself -- owning a pet is like revolting against my family. It's just something that took my fancy.
@Matt (Well, I was partly alluding to that actually.)
Well, there are these issues: (1.) Not every apartment or house allows pets. In fact, most apartments don't. I need to make sure I find one that does. And independent house=more rent, of course.
@Srivatsan: So revolting against your family means that no one else has a pet, I take it? Seeing as you're living alone I would think you don't disturb anyone in your family by having one.
Sure, that's the first step. If both the odd and even terms converge then you are golden -- the series converges. But again this is a sufficient condition, not necessary.
hm, I got that both series diverge. The negative one is -1/(4n) and the other one is 1/(2n)... but I can't sum -oo and +oo, right? Then I can't say anything, can I? ps: thanks already and in advance for your help!
Sure, you're welcome. Let's see why this is not enough to conclude divergence.
Suppose that the sequence is 1/1 - 1/1 + 1/2 - 1/2 + 1/3 - 1/3 + 1/4 - 1/4 + ... Does this sequence converge or diverge? If you separate into odd and even terms, do they converge or diverge?
Ok, the problem with that example is that even though the two series diverged, the positive and negative terms canceled each other perfectly. As you observed, the key to prove divergence is that one of the series grows faster than the other, so that the difference is large numerically (either large and positive or large and negative).
because the series is like -1/4 + 1/2 -1/8 + 1/4 - 1/16 + 1/8 ... and then I can see that 1/4 and 1/8 are gone, but the positive series is always ahead
alright, can I do the radio test?
even though it alternates?
this would divide the odd terms with even terms series
Ratio test might fail, for the same reason it fails for the harmonic series. But you should try it. Can you get back to me after you're done with that?
