Section 1.3 of Burden and Faires: Big O Convergence

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Burden and Faires, 6c, p. 37

Let's consider the sequence

and try to determine its rate of convergence to zero as .

We need to remember those Taylor series polynomials, and think about what's happening as . The argument to sine is getting really small, so sine is approaching 0.

We want to know the rate at which it is approaching zero.

Definition 1.18: Suppose is a sequence which converges to zero, and converges to a number . If with for large , then converges to with rate of convergence .

Therefore

and

Hence we have found a and a that work, and converges to with rate of convergence .



Burden and Faires, 7d, p. 37

Consider the function . We want to find the rate of convergence to -1 as .

Definition 1.19: Suppose that and . If such that for sufficiently small , then .

First we might demonstrate that the limit is, in fact, -1. Use L'Hopital's rule:

Again we use the Taylor series (Maclaurin series, really) to help us out: only in this case we're going to need to go to for :

.

So

Therefore

Hence, for sufficiently small h (), we can choose . Then

, and as .