Do Continuous Functions Contain Convergent Sequences
A fundamental question we can ask about a sequence is whether or not its values tend toward a particular value, just as a continuous function of may or may not approach a limit as
tends toward infinity.
Convergence of Sequences
When we learned about limits of functions, we thought of evaluating the function value for a collection of values that tended toward some specified value
, and tried to find a number that the function values would approach as
became arbitrarily close to
(but did not necessarily reach
). If there was such a number, we called that number the limit of the function at
. If there was no such number (because the function values became infinitely large or continued to oscillate, we said the function failed to have a limit at
.
A similar situation is true for sequences, but instead of letting approach some value
, we let
approach
. In informal language, if the terms of the sequence get closer and closer to some limit
as
gets larger and larger, we say the sequence has a limit
(we have no need to add "at infinity" since the only definition of the limit of a sequence is as
goes to
). We make this more formal with a definition.
When a sequence converges to a limit
, we write
Calculating Sequence Limits
For many sequences, we can use the definition directly to determine whether the sequence converges or diverges and to what limit (we call this the convergence of the sequence). Once we know the convergence of several sequences, we can use these to quickly determine the convergence of sequences with more complicated formulas for the sequence terms. The following rules can be used when the formula for the terms of the sequence can be written as an algebraic sum, product, or quotient of terms of sequences whose convergence is known.
Sequence Limit Theorems
Recall that when we were trying to find the limit of a function, one approach we used was to "sandwich" the function between two other functions that had the same limit, and use a Sandwich Theorem to demonstrate that the limit of the function was the same as the limit of the two sandwiching functions. Since a sequence can be seen as a function that is only defined on the natural numbers, the sandwich theorem should still hold for sequences. We restate the theorem in the language of sequences here.
We can take advantage of the fact that the sequence is a function defined on the natural numbers in another way. If we can define a new function, defined on all real numbers, that passes through the all points of the graph of the sequence, then finding the limit of this new function will give us the limit of the sequence.
When we have a sequence of terms that are fractions for which the numerator and denominator either both have limit 0 or both have infinite limits, we can often apply this theorem to convert the problem into finding a limit of a continous function, then using L'Hôpital's rule to evaluate the limit.
We know that a composition of two continuous functions is continuous, so if we apply a continuous function to the continuous function
that passes through all the sequence points, we will get a new continuous function. Knowing the limit of the sequence gives us the limit of
, which gives us the limit of the composite function, as stated in the following theorem.
Monotonic and Bounded Sequences
We now define some specific types of sequences that have additional properties.
Definition:
If each term (after the first term) in a sequence is greater than or equal to the prior term, we say the sequence is nondecreasing (if each term is greater than the prior term, with no terms equal, we say the sequence is strictly increasing).
If each term (after the first term) in a sequence is less than or equal to the prior term, we say the sequence is nonincreasing (if each term is less than the prior term, with no terms equal, we say the sequence is strictly decreasing).
A sequence that is either nonincreasing or nondecreasing is a monotonic sequence.
There are two common methods to show that a sequence is monotonic. The first is to write a formula for the difference between a term and the prior term, and demonstrate that that difference is either never negative or never positive for all The second is to define a continuous function
with
for all
and showing that the derivative of that function is either never negative or never positive for all
If a sequence is bounded, and is also monotonic, it must increase or decrease forever, but never escape its bounds, which implies that the sequence has a limit somewhere between the upper and lower bounds.
Notice that this theorem does not allow us to compute the limit. It only tells us that a limit exists, and that the limit may not fall outside the bounds for the sequence.
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Source: https://course.math.colostate.edu/calc2-review/lessons/Math.Calc.SeqSer.02.html