Each time you click on the sentence that best describes your problem, more text and/or further questions will appear.
My problem concerns finding the limit of an explicitly given sequence.
th term, or is it defined in some less direct way, such as using a recurrence relation?
th term.
is like?
is a rational function of
, that is, a function of the form
where
and
are both polynomials.
and
then
,
and
if
It helps to divide through by the largest power of
that appears in
. For instance, to prove that
one rewrites it as
and applies the above rules to argue that the top and bottom tend to 1, so the whole fraction tends to 1. It should be noted that this argument freely uses the fact that
as
This is called the Archimedean property of the real numbers.
involves a ratio of two functions that are more complicated than polynomials.
so you want to compare the growth rates of
and
as
tends to infinity. A few techniques can get you quite a long way. For example, if you can find some
and prove that the ratio
is greater than
for every sufficiently large
then
must tend to infinity. This, for example, is enough to show that
tends to infinity. To see this, note that
while
which tends to
as
tends to infinity (because it is a product of 100 sequences, all of which tend to
). Therefore, for sufficiently large
is bigger than
More generally, any exponential function
with
grows faster than any polynomial function
Combining this with techniques for dealing with rational functions will allow you to find limits of sequences such as
involves raising a number to a power that depends on
.
as
then try turning the problem round: if
then
; however, the very simple but surprisingly useful inequality
proves that
when
If is equal to some number close to
raised to a power that depends on
, then you will almost certainly want to make use of the fact that
For example, if you are asked for the limit of
you can argue that it is the cube of
and hence that it tends to
And
is the
th root of
which is less than
(because in fact the sequence
is increasing), so it tends to
(because
tends to
).
involves factorials.
tends to
or to
?
is bounded away from
. For example, if
then
which implies that
Or you may consider using simple estimates for
such as that
which actually follows from the above calculation if we observe that
and
to
tends to 1 as
tends to infinity.
th term
is given by a formula, and that is first to prove that a limit exists, and then to use the recurrence to determine what the limit must be. An example will illustrate the idea. Suppose we define a sequence by taking
and
If we know that this sequence tends to a limit
then both
and
must tend to
(since
also tends to
). But
tends to
, by the rules for adding and dividing limits, so
which implies that
Since every
is positive (by an easy induction),
However, we still need to prove that the sequence converges to something. How can we do this without simply proving that it converges to
? The answer is that there are theorems (or axioms) that say that a sequence with such-and-such a property converges. For example, Cauchy sequences converge. For this particular problem, we use the fact that a monotone decreasing function that is bounded below converges. If we know that
, then it is easy to check that
But
and
is greater than
by the AM-GM inequality applied to
and
Therefore, the result is proved.
I have an explicitly given sequence, and am required to prove that it converges, but I am not required to calculate the limit.
I am supposed to prove rigorously a statement that looks utterly obvious, such as that or
tend to
is monotone decreasing and bounded below by
so it converges to some limit (which must be non-negative). If this limit is
and
then there must be some
such that
which implies that
whenever
But this contradicts the fact that
tends to
A similar argument works for
My problem is to prove that a sequence converges, but rather than being told what the sequence is I just know that it has certain properties.