DinS          Written on 2018/4/27

1. What’s sequence?

Sequence is a list of number with the notation a_n.

We can represent a sequence using two ways.

Formula: a_n = f(x) where . For example a_n = n².

Recursive: a_n = a_n-1 + a_n-2.

 

Two well known types of sequence are arithmetic progression and geometric progression.

An arithmetic progression is a sequence with a common difference between the terms.

General form: a_n = a₀ + d*n.

Example: 1, 3, 5, 7, 9, 11, 13, 15……   (where a₀ = 1, d = 2)

Each term is the arithmetic mean of its neighbors. That’s the reason for its name.

A geometric progression is a sequence with a common ratio between the terms.

General form: a_n = a₀*rⁿ

Example: 1, 2, 4, 8, 16, 32, 64, 128……   (where a₀ = 1, r = 2)

Each term is the geometric mean of its neighbors. That’s the reason for its name.

There’re many trick to deal with these sequences, which I won’t bother to discuss here. You can easily find them on the web or a textbook.

2. The limit of sequence

Sometimes we can work out the limit of a sequence. What does that mean?

First see an example.

Just basic limit calculations.

In plain language,  means that a_n is as close as you want to L provided n is large enough.

As a matter of fact, not all sequence have limit. This should be as clear as day.

Even if the limit exists, it’s usually hard to find it. To get ourselves prepared, we need several concepts.

Bound above : a_n is bounded above means there is a real number M so that for all n>=0, a_n<=M.

Bound below : a_n is bounded below means there is a real number M so that for all n>=0, a_n>=M.

Increasing: a sequence is increasing if whenever m>n, a_m > a_n.

Decreasing: a sequence is decreasing if whenever m>n, a_m < a_n.

Monotone: a sequence is monotone if the sequence is non-increasing or non-decreasing.

These concepts should be clear enough.

3. Monotone convergence theorem

Statement: If a sequence is bounded and monotone, then  exists.

If you try to think about it, it makes sense.

 

Let’s see an example using this theorem.

Given a sequence .

We want to prove that the limit of a_n exists.

First part is to show there’s a boundary. Notice that we don’t need to find the exact boundary. As long as we show that there is a boundary, it’s good enough.

Apparently a_n >= 0. Here we claim that a_n <= 2.

.

Because a₁ = 1, the claim 0 <= a_n <= 2 holds true.

Actually we can use number larger than 2 to prove it.

Second part is to show the sequence is non-decreasing.

Proof: we want to show that a_n <= a_n+1

Now we square both sides and becomes

If we can show the above inequality holds, we’re done.

In fact we can. We’ve already proved that 0 <= a_n <= 2 holds true, which means that each term above is greater than 0. As a result, a_n is non-decreasing.

By monotone convergence theorem the limit of a_n exists.

What’s the limit of a_n? Monotone convergence theorem won’t tell us. 2 is a good guess.