# Infinite Sequences

## Definitions

A sequence of real numbers is a function *f* (*n*), whose domain is the set of positive integers. The values *a*_{n} = *f* (*n*) taken by the function are called the terms of the sequence.

The set of values *a*_{n} = *f* (*n*) is denoted by {*a*_{n}}.

A sequence {*a*_{n}} has the limit *L* if for every *ε* > 0 there exists an integer *N* > 0 such that if *n* ≥ *N*, then |*a*_{n} − *L*| ≤ *ε*. In this case we write:

The sequence {*a*_{n}} has the limit ∞ if for every positive number *M* there is an integer *N* > 0 such that if *n* ≥ *N* then {*a*_{n}} > *M*. In this case we write

If the limit \(\lim\limits_{n \to \infty } {a_n} = L\) exists and *L* is finite, we say that the sequence converges. Otherwise the sequence diverges.

## Squeezing Theorem

Suppose that \(\lim\limits_{n \to \infty } {a_n} = \lim\limits_{n \to \infty } {b_n} = L\) and \(\left\{ {{c_n}} \right\}\) is a sequence such that \({a_n} \le {c_n} \le {b_n}\) for all \(n \gt N,\) where \(N\) is a positive integer. Then

The sequence \(\left\{ {{a_n}} \right\}\) is bounded if there is a number \(M \gt 0\) such that \(\left| {{a_n}} \right| \le M\) for every positive \(n.\)

Every convergent sequence is bounded. Every unbounded sequence is divergent.

The sequence \(\left\{ {{a_n}} \right\}\) is monotone increasing if \({a_n} \le {a_{n + 1}}\) for every \(n \ge 1.\) Similarly, the sequence \(\left\{ {{a_n}} \right\}\) is called monotone decreasing if \({a_n} \ge {a_{n + 1}}\) for every \(n \ge 1.\) The sequence \(\left\{ {{a_n}} \right\}\) is called monotonic if it is either monotone increasing or monotone decreasing.

## Solved Problems

### Example 1.

Write a formula for the \(n\)th term of \({a_n}\) of the sequence and determine its limit (if it exists).

Solution.

Here \({a_n} = {\frac{n}{{n + 2}}}.\) Then the limit is

Thus, the sequence converges to \(1.\)

### Example 2.

Write a formula for the \(n\)th term of \({a_n}\) of the sequence and determine its limit (if it exists).

Solution.

We easily can see that \(n\)th term of the sequence is given by the formula \({a_n} = {\frac{{{{\left( { - 1} \right)}^{n - 1}}n}}{{{2^{n - 1}}}}}.\) Since \( - n \le {\left( { - 1} \right)^{n - 1}}n \le n,\) we can write:

Using L'Hopital's rule, we obtain

Hence, by the squeezing theorem, the limit of the initial sequence is