A sequence is convergent if there is a number L such that if k \ \rightarrow \ \infty, then t_{k} \ \rightarrow \ L . Mathematically it is represented as follows:
\underset{k \to \infty} \lim \; \ t_{k} \ = L
What the previous expression means is that if k is infinite, then there will be infinite positions, but each of them will always be equal to a number.
On the other hand, a sequence is divergent when its limit is infinity.
To explain it more simply, the Cauchy criterion is used, which tells us that a sequence is convergent if, given an arbitrarily small number \varepsilon, there exists a number N>0 such that:
\left| t_{p} - t_{q} \right| \ \text{ for } \ p,q>N \ \text{ y } \ q = p+1
In other words, if \left|t_{p} - t_{q}\right|< \varepsilon, then it is convergent. If it turns out that \left|t_{p} - t_{q}\right|> \varepsilon, then the sequence is not convergent.
Let’s see some examples when a sequence is convergent and when it is not convergent and we will take a \varepsilon that is less than 0.003.
Example of convergent sequence
We have the sequence: t_{k} = \cfrac{n-1}{n}. We will prepare the following table:
\begin{array}{c c c}
k & t_{k} & \left|t_{p}-t_{q}\right|\\
1 & 0 &\\
2 & 0\text{.}1250 & 0\text{.}1250 \\
3 & 0\text{.}1667 & 0\text{.}0416 \\
4 & 0\text{.}1875 & 0\text{.}2083 \\
5 & 0\text{.}2000 & 0\text{.}0125 \\
6 & 0\text{.}2083 & 0\text{.}0083 \\
7 & 0\text{.}2142 & 0\text{.}0059 \\
8 & 0\text{.}2187 & 0\text{.}0044 \\
9 & 0\text{.}2222 & 0\text{.}3400 \\
10 & 0\text{.}2250 & 0\text{.}0027
\end{array}As you can see, when t\to \infty, then \left|t_{p} - t_{q}\right| \to 0, therefore, since the difference of the absolute value of t_{p} and t_{q} is less than \varepsilon, then the sequence does converge.
Non-convergent sequence example
We have the sequence t_{k} = t_{k-1} + 2 with t_{1}=1. We will prepare the following table:
\begin{array}{c c c} k & t_{k} & \left|t_{p}-t_{q}\right|\\ 1 & 1 & \\ 2 & 3 & 2 \\ 3 & 5 & 2 \\ 4 & 7 & 2 \\ 5 & 9 & 2 \end{array}When k\to \infty, then \left | t_{p} - t_{q} \right|> \varepsilon; therefore the moment will never come when \left| t_{p} - t_{q} \right| < \varepsilon, so the sequence is not convergent.
We already know when a sequence is convergent and not convergent, let’s go out into the world for those sequences!
Thank you for being in this moment with us : )