11th Grade - Sequences and induction

5 - Sequences and induction

Vocabulary

You've seen in the 10th Grade lesson about the sequences what an increasing sequence is, a decreasing sequence, monotonic, bounded above, below, and both. Now, let's see what a convergent sequence is and what are adjacent sequences.

A convergent sequence approaches a certain number, called limit of the sequence, when n tends to infinity. It is a sequence u such that there is a real number l such that .

Two adjacent sequences are such that one is increasing and the other decreasing and the terms of both are getting closer as n tends to infinity, that means .

Examples :
- the sequence defined for every n by is decreasing, monotonic, bounded above and below and convergent. Its limit is 2.
- the sequence defined for every n by is bounded and divergent.

Notice that an increasing sequence is always bounded below by its first term, and a decreasing sequence is always bounded above by its first term. A monotonic sequence can be either convergent or divergent.

Properties

Every bounded above and increasing sequence is convergent and every decreasing and bounded below sequence is convergent (but be careful, their limit is not always equal to the majorant or minorant).
If two sequences are adjacent, then they are convergent and converge to the same limit.

Sequences defined by induction

A sequence is defined by induction when you are given the value of one term and a relation between the general term of order n and the next term of order n+1. For example, the sequence is defined by induction. f is the function that gives in function of . If you know that the sequence u is convergent and the function f is continuous in l , then, by passing to the limit in the induction relation, we obtain the equality . Generally, this equation enables us to calculate l.

Also, note that for two sequences defined this way, you can determine an approximate value of the terms of the sequence and make conjectures about the convergence of the sequence, thanks to a drawing. Then let's draw in an orthornormal frame the graph of f, and let's place on the X-axis the first term . We have then using the graph of f, let's place on the Y-axis the term . Now let's draw the line of equation equation y=x. Come back, from on this line, and go down on the X-axis, so that you put back on the X-axis. Now, using the graph of f, place on the Y-axis and put back its value on the X-axis thanks to the line y=x. This way you can place as many terms of the sequence on the X-axis as you want and guess the limit of the sequence.

Mathematical induction

Mathematical induction is a method of reasoning that enables us to prove that some property is true for every number n. For example, if you have to prove that is always a multiple of 3, you will generally use induction. Induction consits in 4 steps.

1. Suppose ="the property you want to prove", for example here you'll suppose

2. Show that is true. Generally it's rather simple. Here is true because and 0 is a multiple of 3.

3. Show that for every number n, if is true, then is also true. It's the most difficult step. To write the solution, you will for example say : "n is a natural number. Suppose that is true.". We want to show that is also true, that means that is a multiple of 3.

is a multiple of 3.
is a multiple of 3 because is true. The sum of two multiples of 3 is a multiple of 3, hence is a multiple of 3, thus is a multiple of 3, and therefore is true.

4. Then conclude. Since is true, and since for every n, , we have , then is true, then is true... and therefore is true for every n. To say it, just write : "using the induction principle, is true for every n".

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Mathematical induction

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