How To Find If A Sequence Converges

Imagine a Kung Fu blackness belt took a function and chopped through information technology, leaving only detached values. Those discrete values would form a sequence. Because the sequence is only a coarse-chopped list of numbers fabricated from a function, the sequence acts in means like to functions.

Let's look at some of the ways the martial arts master can serve up a sequence to the states. These meals are going to look like to functions, so you may want to review it to run across the similarities.

We say a sequence is increasing if the terms get larger every bit n gets larger. This is like a lop-sided sushi roll where the piece on the right is bigger than the one to its left.

In symbols, if thou < north then a₁₀₀₀ < a_n .

We say a sequence is decreasing if the terms get smaller as n gets larger. This is like a lop-sided sushi roll where the piece on the correct is smaller than the one to its left.

In symbols, if yard < n the a_m > a_north .

If a sequence is increasing or decreasing we call it monotonic because the terms are going only 1 mode. Nosotros'll have a monotonically increasing California curl, extra wasabi on the side.

Sample Problem

The sequence a_{due north} = north is increasing because the terms get larger as northward gets larger.

Sample Problem

The sequence is decreasing because the terms get smaller every bit north gets larger.

Exist Careful: increasing and decreasing aren't opposites. It's possible for a sequence to be neither increasing nor decreasing.

Sample Problem

The terms of the sequence a_n = (-i) ^{due north} bounce back and forth betwixt 1 and -1. This sequence is neither increasing nor decreasing.

Be Careful: Using the word "increasing" to refer to a part is ambiguous because it could mean either nondecreasing or strictly increasing. We don't usually intendance almost nondecreasing sequences. They are nigh equally interesting as watching water evaporate off a hot road surface in the eye of summer. That'due south why, for sequences, we utilize "increasing" as an abbreviation for "strictly increasing".

Simply similar a function, nosotros say a sequence is bounded to a higher place if all terms of the sequence are less than or equal to some value Thou.

In symbols,

a_n ≤ Chiliad

for all n.

No surprises here. Nosotros say a sequence is divisional beneath if there's a value Yard such that all terms of the sequence are at least K.

In symbols,

a_n ≥ K

for all n.

If a sequence is bounded in a higher place and below, we say it'southward bounded. For our sushi sequence, if it is bounded, we tin can make a bento box with it.

If a sequence is missing one or both of these bounds, and then it'southward unbounded.

There are a couple of theorems connecting the ideas of boundedness and convergence for sequences. These are some of the ideas that spice up our sushi roll sequences.

Theorem

A monotonic divisional sequence must converge. This is a pretty obvious statement, so we could phone call this the california roll theorem. Anybody knows what it is, and as boring as it may exist, everyone eats it.

Proof. Rather than be overly mathematical, we'll explain things out to give yous the idea. If a sequence is increasing, the terms are going upward.

If the sequence is bounded, the terms can't go upwardly forever, considering they can't go higher up the upper bound.

That means the sequence converges. If the upper bound given wasn't the best upper bound possible, the sequence could converge to some value L smaller than the given upper bound.

Similarly, if the sequence is decreasing the terms are going down. If the sequence is bounded, the terms can't go downwardly forever, then they must approach some flooring. That means the sequence converges.

Theorem

Any finite sequence is divisional.

Proof. A finite sequence has some largest term and some smallest term. These give upper and lower premises, respectively.

Theorem

A convergent sequence must be bounded. We could call this the bento box theorem. If the roll converges to some size, information technology volition fit in a box.

Proof. If a sequence converges to some value L, so eventually all the terms must be very shut to L. In particular, eventually the terms must exist within i of 50 in either direction.

Formally, when n gets large enough we have

L – 1 ≤ a_n ≤ L + 1.

There can exist but finitely many terms, all at the beginning of the sequence, that aren't within 1 of L.

Have the largest term that isn't within one of L. If this term is bigger than Fifty + 1, this term is an upper bound for the sequence.

Otherwise, L + one is an upper jump for the sequence.

Similarly, accept the smallest term that isn't within 1 of 50. If this term is smaller than L – one, this term is a lower bound for the sequence.

Otherwise, Fifty – 1 is a lower bound for the sequence.

In dissimilarity, a bounded sequence does Not have to converge. This non-theorem is like a dragon ringlet. It will always keep y'all on your toes. One of the easiest examples is the sequence

a_n = (-1) ⁿ .

This sequence is definitely divisional, since -one ≤ a_northward ≤ 1 for all northward. However, the sequence can't converge considering it'due south indecisive.

Source: https://www.shmoop.com/study-guides/math/sequences/convergence-divergence-sequence

Posted by: davisexter1987.blogspot.com

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