Tom: Hey Satan...

Satan: Yeah?

Tom: I made a graph.

Satan: A what?

Tom: You know, a graph, you fucker. It illustrates my current hypothesis that happiness is impossible.

Satan: Oh Christ, lemme see it.

Tom: Sure...



Tom: See, for there to be such a thing as "happiness" there would have to be a point in the space-time continuum at which one stops being "sad" and becomes consciously "happy". But how can this happen? Can one ever be completely happy? Can one ever be completely sad? Of course not. We are lead to believe that if we only ate less, exercised more, played more X-treme sports, and just tried hard enough, everything would be SWEET SHIT.

Satan: Hmmm...

Tom: Which leads me to ask the question "Who is the asshole that spreads this nonsense?" At best it's just bad semantics! At worst it's a fallacy, a lie, a sick joke sold to us by conniving faith healers, intent on controlling our emotions and making us feel responsible for the SHITTY things that happen to us. I for one will not stand for it! What do you think of that, bitch?

Satan: I don't know...it sounds to me like you've stumbled onto one of Zeno's paradoxes.

Tom: Come again?

Satan: Zeno's paradoxes are a set of paradoxes conceived by Zeno of Elea to support Parmenides's doctrine that all evidence of the senses is misleading, and particularly that there is no motion.

Tom: Yeahhh...?

Satan: Well, in the paradox of Achilles and the tortoise, we imagine the Greek hero Achilles in a footrace with a stupid reptile. Since Achilles habitually indulged in an ancient form of methamphetamine known to the Greeks as crankus, which allowed him to perform superhuman feats over an extended period of time, Achilles graciously gives the tortoise a head start of a hundred feet.

Tom: I don't give a shit...

Satan: If we suppose that each racer start running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run a hundred feet, bringing him to the tortoise's starting point; during this time, the tortoise has "run" a (much shorter) distance, say one foot. It will then take Achilles some further period of time to run that distance, during which the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, Zeno says, Achilles can never overtake the tortoise.

Tom: So...?

Satan: So therefore, by your reasoning, one can never be happy because as one approaches your theoretical "brink of happiness" it too continues to move along in the space-time continuum, always just ahead of us, always eluding us no matter how fast we (you) careen towards it.

Tom: See? I'm right. Suck my dick.

Satan: Sorry to tell you, but the paradox is resolved with the fundamental insight of calculus that a sum of infinitely many terms can yield a finite result. Adding the (infinitely many) times together that Achilles needs to reach the previous positions of the tortoise results in a finite total time, and that is indeed the time when Achilles overtakes the tortoise.

Tom: What the fuck are you talking about?

Satan: An infinite series is a sum of infinitely many terms. Such a sum can have a finite value, and if it has, it is said to converge. The fact that infinite series can converge resolves several of Zeno's paradoxes. Here, let me draw it out for you:

The simplest convergent infinite series is perhaps

1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2

Tom: Bullshit.

Satan: I think not. It is possible to "visualize" its convergence on the real number line. This series is a geometric series and mathematicians usually write it as:



Formally, if an infinite series:



is given with real (or complex) numbers an, we say that the series 'converges towards S ' or that its 'value is S ' if the limit:



exists and is equal to S. If this is not the case, we say the series diverges. How's about them apples, ya fucker?

Tom: We'll get back to this, cockface. I gotta go to school.