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
.