I can't understand anything about infinity without limits |
That's good, because in calculus all infinities come with limits.
There is nothing particularly complex about a plain converging sequence. The idea is that you can find a tail which is as close to some point as you'd like. Take a sheet of paper, choose a limit point C, then put a point anywhere on the sheet at most 1 cm away from C, that's A
1. Then put a point anywhere on the sheet at most 1/2 cm away from C, that's A
2. If you continue, you'll make a sequence {A
n} that converges to C. In fact, if you instead put multiple points at each distance level, you could make all sequences that converge to C that way. Of course, in calculus, you work in a line, rather than a plane. That only makes thing simpler though.
A series converges iff the sequence of its partial sums converges. You can construct any series by taking some sequence and summing over A
i-A
i-1. It's only the presence of summation operator that makes series harder to work with than plain sequences.
An integral converges if you can give lower and upper approximations to the area under a curve that are as close to each other as you'd like. Riemann made his approximations by drawing thin rectangles, so you will too. While it could be reduced to a sequence, a proper integral is fairly nice so you don't have to actually look at it as a limit of two sequences.
An impropper integral is a limit of proper integrals that approach the "bad" point. The integral of 1/x
2 converges because the sum of 1/n
2 i.e. a bunch of rectangles 1 unit wide and 1/n
2 units high, converges (though it's more often explained the other way).