Convergent Sequences

A sequence a_n is said to be convergent if its limit as n→∞ is a finite number l: $$ \lim_{n \rightarrow \infty} a_n = l $$ In other words, for any ε > 0, there exists some number v such that: $$ l - \epsilon < a_n < l + \epsilon \quad \forall n > v $$
Equivalently, this condition can be expressed as: $$ | a_n - l | < \epsilon \quad \forall n > v $$

Every convergent sequence is also a bounded sequence.

However, the converse is not necessarily true: a bounded sequence is not always convergent.

For instance, the sequence (-1)ⁿ oscillates between -1 and +1. It is bounded but does not converge.

Note. If a sequence converges to zero, it’s referred to as an infinitesimal sequence. This is a special case of a convergent sequence. $$ \lim_{ n \rightarrow \infty } a_n = 0 $$

A Practical Example

Consider the sequence:

$$ a_n = \frac{n-1}{n} $$

The limit of this sequence as n→∞ is one:

$$ \lim_{ n \rightarrow \infty } \frac{n-1}{n} = 1 $$

Therefore, the sequence is convergent because its limit as n→∞ is finite.

Verification

A sequence is convergent if, for every ε > 0, the following inequality holds:

$$ | a_n - l | < \epsilon $$

In this example, l = 1:

$$ | a_n - 1 | < \epsilon $$

The general term a_n is defined as (n-1)/n:

$$ \left| \frac{n-1}{n} - 1 \right| < \epsilon $$

Let’s simplify this expression through a few algebraic steps:

$$ \left| \frac{n-1 - n}{n} \right| < \epsilon $$

$$ \left| \frac{-1}{n} \right| < \epsilon $$

Since n > 0, we can drop the absolute value signs:

$$ \frac{1}{n} < \epsilon $$

Rearranging gives us:

$$ n > \frac{1}{\epsilon} $$

This shows that if the index n exceeds 1/ε, the difference |a_n - l| will indeed be less than ε.

Therefore, the threshold value v is equal to 1/ε:

$$ v = \frac{1}{\epsilon} $$

For all n > v, the condition |a_n - l| < ε is satisfied.

This confirms that the sequence is convergent.

Note. In the graph above, ε = 0.5, so v = 1/ε equals v = 1/0.5 = 2. For every index n > v, i.e., n > 2, the sequence a_n lies entirely within the neighborhood (l - ε, l + ε).

And so on.