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      Calculus Methods

      Sequences

      Sequences are to calculus what at calculator is to a scientist. There are many ways to introduce sequences. Here we will follow a somewhat unorthodox way. Indeed, consider a scientist doing an experiment; he is collecting data, let us say, every day. So, put to be the data collected the first day, be the data collected the second day, and so on. . . . and is the data collected after n days. Clearly, we are generating a set of numbers with a very special characteristic: there is an order on the number, that is, we naturally have the first number, the second number, and so on. . . . A sequence is by definition a set of real numbers with this natural order. We will use the notation

      {X_n}_n≥1,

      to describe the sequence of numbers where X_n is the nth number.

      Definition (Range):Consider the sequence {X_n}_n≥1,. The set

      {x₁, x₂, x₃, . . . .} = {x_n; n = 1, 2, 3, . . .}

      is called the range of the sequence.

      Of course, in the range there is no order. For example, consider the sequence {(–1)ⁿ}n≥1 Its range is the set {–1, 1}. It has two elements. The sequence itself is alternating between 1 and -1.

      Series

      The notion of series is closely related to the sum of numbers. In fact, whenever one hears the word series, the first thing to come to mind is the sum of numbers. This is the basic difference between series and sequences. So series, as we will see, are here to help us add numbers. So what is the problem? Let us do a simple addition problem. Let me give 3 numbers: A, B and C. If I ask you to add them, you will take a pen and a paper or a calculator and do the following:

      1. You will enter the number A, then

      2. You will add to it the number B, to get A+B

      3. And finally, you will add C to the previous result to get A+B+C

      So, if you are given one million numbers, you will still enter one by one to add all of them. A very simple operation, isn’t it? Now what would happen if you were given infinitely many numbers? What would you do?

      Example: Add the numbers

      Answer: First we enter (into our calculator) the number ½. The output will be