A calculus of the absurd

Chapter 19 Real analysis

19.1 Sequences

A sequence is a lot of numbers, one after the other. Usually the terms are related in some way to the previous terms. For example, in this sequence136136 Known as the Fibonacci sequence, each term is the sum of the two previous terms137137 If we set that the first term is \(1\) and the second \(2\) then to get the third term, we add the previous two, so \(1 + 2 = 3\). This process continues for the rest of the terms in the sequence.

\begin{equation} 1, 2, 3, 5, 8, 13,... \end{equation}

We can also write the nth term of a sequence \(x\) as \(x(n)\), and denote the entire sequence as \(\{x_n\}\)

19.1.1 Convergence of sequences

Sometimes a sequence can converge, which informally means that the value of the \(n\)th term in the sequence gets closer and closer to a specific value - infinitely close in fact!

Can we say more than “this sequence looks like it’s getting closer and closer to this value” though? Yes! Another way of saying that something is getting “closer” to a value, is to say that the distance between the two is decreasing. Using the modulus function 138138 There is a section on the modulus function in the "Algebra" chapter of this document, and another section on how we can write distances in terms of the modulus function in the vectors chapter. we can write the distance between any two points \(x(n)\) and \(a\) as

\begin{equation} \abs {a(n) - a} \end{equation}

What about "really close"? The way to do this is to say that for every value of \(\epsilon > 0\) where \(\epsilon \) is a real number and as \(n \to \infty \) we have that

\begin{equation} \abs {x(n) - a} < \epsilon \end{equation}

then \(\{x_n\}\) converges to \(a\). As we can pick any \(\epsilon > 0\) (but not \(\epsilon = 0\)) what we are in effect saying is that the distance is infinitely close to zero, but not equal to zero. That is, it converges to \(0\) but is not equal to zero.