Consider the convergent sequence with terms defined by
Let . Find the value of
and values of
such that
for all
for each of the following values of
:
-
,
-
,
-
,
-
,
-
.
First, we know
So then we have,
Thus, if then for every
we have
. We compute for the given values of
as follows:
-
implies
.
-
implies
.
-
implies
.
-
implies
.
-
implies
.
I find myself nitpicking here: When you passed to the absolute value you must have assumed the limit L is zero, which is obvious in this case. But beyond knowing the limit in advance Apostol didn’t give any tools for actually determining convergence. What little I know of analysis it seems something like all convergent subsequences sharing the same limit would work, or perhaps cauchy criterion? It just seems that Apostols exposition of sequences was less then comforting; but maybe it will be covered later.
Greetings!
There are of course alternative to solve this exercise. You can take
and show that for
we indeed have
for
arbitrary. Thus proving
as
using only the definition of convergence. From
you can also create the required table of
. (I use
instead of
).
Yeah but it should be
?