I hope by "one of them has a larger value at every point in its domain" that each partial sum of the first series (w.l.o.g.) is larger than the corresponding (=same index) partial sum of the second series, and I also assume that you're talking about absolute convergence, because else your question wouldn't really make sense..

Now, assuming that we're in R, take a look at the difference between each partial sum of series 1 and series 2, which is strictly positive and converges to 0 - this is basically 'it': One sum is larger than the other, but the difference between the partial sums vanishes in the limit.

Hope that was helpful, I'm usually not good at explaining things, but thought I might give it a try.

>>169
I think he means that the elements of one sequence are bigger at any given index. As in, a_i > b_i. All the same, I'm pretty sure it can work out okay.

Well, are the values of the "larger" series greater by a diminishing amount? Because as the difference between the two terms converges to zero, the partial sums converge to the same value.

>>something proven to be smaller adding up to be the same size of a larger object.

Smaller +smaller =bigger

>>331

I think you're missing the point, what I meant is better expressed by >>170 and >>169.

So, is 99chan taking any interesting math classes this semester? I'm in Abstract Algebra and a class called Foundations of Set Theory at the moment. In the fall I'll probably take some logic related classes, I've heard really interesting things about a newer sub-field called "reverse mathematics".

The text on it is probably best suited to an early graduate student (above me, it's my goal to make sense of it), but "Subsystems of Second Order Arithmetic" is apparently the best on the subject at the moment from what I've heard.

>>333
I'm complex analysis and category theory next semester, which I'm extremely excited for.
Reverse mathematics sounds really cool.

>>333
Supposedly a higher-level approach to probability, which I'm excited for, since I failed the fuck out of stats (after completing the calc series and an intro diff eq, and intro linear alg) but really want to love it. I just can't memorize worth a damn.

>>335

Memorization shouldn't be too important in the higher level probability course, hopefully it is more proof based rather than rote mechanical slavery. Though I haven't taken any probability courses so I may be wrong.

The arts and computational sciences is like communing with the minds of other humans, while writing proofs is directly communing with the mind of God.