How a Sequence Converges

Convergence of Sequence

Under the guidance of Prof. Gaurangadeb Chattopadhyay, Department of Statistics, University of Calcutta, I do this interactive work, for making our study much more interesting. We almost everyone familiar with the definition and results of convergence of a sequence, but it becomes much interesting as we start to visualize them. The usefulness of \(\epsilon\) becomes so clear after that.
Definition. A sequence \(\{a_n\}\) is said to be convergent if, for every \(\epsilon>0\) there exist a natural number \(n_o \in \mathbb{N}\) such that,
\[|a_n -l|<\epsilon \text{ whenever } n \geq n_0\]

Example. Consider a sequence, \[a_n = (1+\frac{1}{n})^n\] From our theoretical practice we know that \[(1+\frac{1}{n})^n \rightarrow e \text{ as } n \rightarrow \infty \]

Here we want to visualize the \(n_0\) as per our example.
We know that \(n_0\) depends on the chosen \(\epsilon\). As we take \(\epsilon\) low, the value of \(n_0\) will increase accordingly. And the sequence become bounded whenever \(n\geq n_0\).

For this we created a R shiny web app for better experience. Lets see that

Link for the shiny app : My shiny app