Orders of Infinity

Excerpts

Orders of Infinity
In the language of Leibnitz if $h$ be an infinitely small quantity, (1) is an infinitely small quantity of the first order, (2) is an infinitely small quantity of the second order, and so on.
Orders of Infinity
Hence (1) is said to be *comparable* to the first power of $h$, or *of the first order*, since this is the only power of $h$ whose ratio to (1) tends towards a finite limit.
Orders of Infinity
As $h$ is diminished, all these expressions decrease without limit; but the first *increases* with respect to the second, that is, contains it more times after a decrease of $h$ than it did before.
Orders of Infinity
Nevertheless this decrease increases the ratio of the first to the second, of the second to the third, and so on, and the increase is without limit.
Orders of Infinity
The converse proposition is readily shown, that if the ratio of two series arranged in powers of $h$ continually approaches to some finite limit as $h$ is diminished, the two series are of the same order, or the exponent of the lowest power of $h$ is the same in both.

Orders of Infinity
\phi' x\, dx + \phi'' x\, \dfrac{(dx)^{2}}{2}
The increment QP' along the curve, to the first two orders in dx, is the first derivative of phi at x times dx plus the second derivative times half (dx) squared, with further terms omitted.

No exercises in this chapter.