Today’s soundtrack is Kendrick Lamar: Damn, a rap album with mesmerizing production and fascinating lyrics.
This morning, I’m moving on to the second preliminary chapter of Calculus Made Easy, “What is a Limit?”
Derivatives and integrals, says Gardner, the fundamental concepts of differential and integral calculus, are both limits. Thus, an understanding of limits is important when studying calculus.
A sum of a finite series is a limit. If values in the series cannot pass x, then the sum of its values cannot pass y, its limit. An asymptote is also a limit. We can get ever closer to it but can never pass it. The numbers converge on the asymptote. But, notes Gardner, “[i]f there is no convergence, the series is said to ‘diverge'” (Thompson, 1998, p. 19).
Can a limit ever be reached? Yes, says Gardner. 1/2 + 1/4 + 1/16… = 1. “The halving series,” he says, “[…] converges exactly on the limit” (Thompson, 1998, p. 20). In fact, there are three kinds of limits:
- The partial sums get ever closer to the limit without actually reaching it, but they never go beyond the limit.
- The partial sums reach the limit.
- The partial sums go beyond the limit before they converge.
(Thompson, 1998, p. 19)
So an asymptote is an example of the first. The halving series is an example of the second. But what of the third? How can anything go beyond a limit if it is truly a limit? We can do this “by changing every other sign in the halving series to a minus sign: 1/2 – 1/4 + 1/8 – 1/16+ . . . . The partial sums of this ‘alternating series’ are alternately above and below the limit of 1/3” (Thompson, 1998, p. 21).
Another way of thinking of a limit is that the closer we get to a limit, we reach values that get ever smaller and smaller, finally reaching the point of being “infinitesimal” – values so small that Martin quotes Johann Bernoulli as saying that “‘if a quantity is increased or decreased by an infinitesimal, then that quantity is neither increased or decreased'” (Thompson, 1998, p. 22). But if you add enough infinitesimals, you will slowly progress. This is why, given enough time, the sum of the harmonic series (1/1 + 1/2 + 1/3 + …) will never actually reach a limit.
That’s it for today! Next time, we’ll be learning about derivatives.