# Section 1.3 of Burden and Faires: Big O Convergence

### From www.norsemathology.org

## 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 α. If with
for large *n*, then
converges to α with **rate of convergence** *O*(β_{n}).

Therefore

and

Hence we have found a and a that work, and converges to 0 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 *h*, 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 .