Математика. Материалы курса

This lecture is devoted to the most significant concept of mathematical analysis, a numeric sequence.

A numeric sequence is a numeric function whose domain is the set of natural numbers.

What does it mean? Each natural number corresponds to an unambiguously defined real number. Let n be a natural number, and we denote the number that corresponds to it with the letter a with the index n. In this case, the numeric sequence has the symbol a with the index n in parentheses. Here is the symbol for the numeric sequence.

If we write curly brackets, this means a set of values for this function; the set of values of a numeric sequence. Sometimes this symbol represents a numeric sequence. However, we are going to use it for the set of values of a numeric function in our course.

A simple example. A numeric sequence always contains infinitely many elements, and this differs it from a set. Each natural number corresponds to a real number. This is the а_n sequence: -1, 1, -1, 1, -1, 1 and so on. There are infinitely many terms. The first term is -1, the second is 1, the third is -1, and the fourth is 1. The sequence number corresponds to the index number n. As for the set of values, there are only two numbers -1 and 1 there. The set of values is a two-element set.

Methods for setting a numeric sequence.

The analytical method similar to the function setting. It is the general term assignment formula. We specify the formula used to calculate the sequence element for each n. Here is an example. Let's substitute n for 1, 2, 3, and so on. We get real numbers. We write them in a row separated by commas, and the ordinal number is the n number for which this number is found.

A new way that we cannot apply for functions. The recurrence relation sometimes sets a numeric sequence. The word "recursion" means getting back. We have to get back to something. Let's consider some examples. Example 1. The first term of the sequence is 3. What about each n + 1? We need to take а_n - the previous one and add 2. Thus, а₂ is obtained if we add 2 to а₁, 3 + 2 is 5, then we add 2 to 5, and we get 7, we add 2 to seven, and get 9, then 11, 13, and so on. This is the recurrence relation. In order to use it, you need to specify one or more of the first members of the sequence, or a recurrence relation that allows you to find the next members.

Example 2. It is very interesting. а₁, а₂ are set, these are two units. Each member of the sequence is equal to the sum of the previous two. If the first two numbers are 1, 1, the next one is 2 (this is one plus one), then we add 1 plus 2, and get 3, then 2 plus 3, 5, 7, and so on. This is a significant, rather interesting sequence, the Fibonacci sequence. It is associated with very interesting sections: the golden section, for example.

The descriptive method. There is no formula; however, we can easily get the sequence. For example, while pronouncing the natural numbers (one, two, three, and so on) we count the number of letters in these words. Then the number of letters in the word one is three, the number of letters in the word two is three, and so on. We get a numeric sequence. We don't even need to try to set the formula here. This is a descriptive way.

How is the numeric sequence represented? Most often, we represent a sequence as points on a numeric line. Let's consider the same example. a_n =1 / n. The numbers 1, ½, 1/3 ... What do we do? We mark the first element on the number line, we write that it is one, а₁. Then 1/2 is а₂, 1/3 is а₃, and so on, because the same image can correspond to different sequences.

The second method of presenting is using points in the coordinate plane. A numeric sequence is a function, and we build a graph for a numeric function. The graph of a numeric sequence consists of n points; 1/n points for a given sequence. We build a coordinate plane, and mark the specified points for each n. We get an image of a numeric sequence.

Properties of a numeric sequence. Since this is a function, we are going to come across some of the function properties, although the sequence is presented differently. There are some specifics here.

A sequence is top-bounded if no members exceed any real number. The logical form of writing the definition is on the slide, and we have just said how to present it in words.

When we represent a sequence by points of a numeric line, bounding from above means that there is a number M on the line and there are no points of this sequence to the right of this number. Here is a simple example. а_n = - n is the formula for the general term of the sequence. – 1, - 2, - 3... If we take 0 as M, there are no points in this sequence to the right of 0. Note that this M is defined ambiguously. We can take 2, or we can take -1, and there are no members of this sequence to the right of -1. The main thing is that there is such a number.

The lower bound is defined similarly. There is some number m and there are no sequence points to the left of it. In other words, on the number line there is a point m, to the left of which there are no points in the sequence.

Let us consider an example. The sequence is set by the formula а _n = n, that is, the numbers 1, 2, 3... are all natural numbers. We see that there are no points in this sequence to the left of zero. We can take as m 1, 0.5, -2, and so on. There are infinitely many values of m. The main thing is that we can find it.

Just like for a function, we introduce the definition of boundedness. A sequence is bounded if it is bounded at the top and bottom. There are two equivalent definitions, just like for a function. We use these definitions to prove that the sequence is limited.

We can define monotonicity in the same way as for a function, but given the specifics of the sequence, we introduce the following definition.

A sequence is increasing if for any n а_n is less than or equal to а _{n + 1}. Here is an example of this sequence: 1, 2, 2, 3, 3, 3, 4, 4, 4, 4. We see that each subsequent member of the sequence is greater than or equal to the previous one. This is an example of an increasing sequence.

Strict increasing means that each subsequent term is strictly greater than the previous one. Similarly, we define strict descending.

Functions can be constant. As for the sequence, is is called stationary if all its members are equal among themselves.

The stationary sequence. Let us consider an example: we need to prove that the sequence with a common term а_n=2n + 1 is strictly increasing. If we need to prove strict increasing, then we write down the definition of a strictly increasing sequence. What should we do? Let's take an arbitrary n and prove that а_n is less than а_{n + 1}. The condition is that а_n is given: 2n +1. To prove that а_n is less than a_{n + 1}, we calculate a_{n + 1} using the general term formula. We get 2n + 3. Obviously, this is more than 2 n + 1. That is, а_n is strictly less than а_{n + 1} by the definition of strict increasing. The proof is complete.