Limit of an Arithmetic Sequence

The behavior of an arithmetic sequence $ a_n = a_1 + (n-1)d $, as the index $ n $ tends to infinity, is determined by the value of the common difference $ d $.

If $ d = 0 $, the sequence is constant, and its limit coincides with the first term: \[ \lim_{n \to +\infty} a_n = a_1 \]
If $ d \neq 0 $, the sequence does not converge but diverges to infinity: \[ \lim_{n \to +\infty} a_n = \begin{cases} +\infty & \text{if } d > 0 \\ -\infty & \text{if } d < 0 \end{cases} \]

Equivalently, an arithmetic sequence has a finite limit if and only if it is constant; otherwise, it diverges.

An arithmetic sequence is a sequence of numbers in which consecutive terms differ by a constant quantity, called the common difference and denoted by $ d $.

Each term can be defined recursively by adding the common difference to the preceding term:

$$ a_n = a_{n-1} + d $$

An explicit expression for the general term is given by:

\[ a_n = a_1 + (n-1)d \]

This formula makes it straightforward to analyze the behavior of the sequence as $ n \to +\infty $.

There are two cases to consider.

1] Case 1: the common difference is zero

If $ d = 0 $, every term is equal to the first term, so the sequence is constant.

It follows immediately that:

\[ \lim_{n \to +\infty} a_n = a_1 \]

2] Case 2: the common difference is nonzero

If $ d \neq 0 $, each term is obtained by repeatedly adding or subtracting the same fixed amount. Consequently, the sequence is unbounded.

If $ d > 0 $, the sequence is strictly increasing and diverges to $ +\infty $.
If $ d < 0 $, the sequence is strictly decreasing and diverges to $ -\infty $.

In compact form:

\[ \lim_{n \to +\infty} a_n = \begin{cases} +\infty & \text{if } d > 0 \\ -\infty & \text{if } d < 0 \end{cases} \]

Therefore, every arithmetic sequence with $ d \neq 0 $ is divergent.

Examples

Consider an arithmetic sequence with initial term $ a_1 = 5 $ and common difference $ d = 0 $.

The sequence is:

\[ 5, 5, 5, 5, \dots \]

Since the sequence is constant, we have:

\[ \lim_{n \to +\infty} a_n = 5 \]

Example 2

Consider an arithmetic sequence with initial term $ a_1 = 2 $ and positive common difference $ d = 3 $.

The sequence is:

\[ 2, 5, 8, 11, 14, \dots \]

The sequence is strictly increasing and unbounded above. Hence:

\[ \lim_{n \to +\infty} a_n = +\infty \]

Example 3

Consider an arithmetic sequence with initial term $ a_1 = 10 $ and negative common difference $ d = -2 $.

The sequence is:

\[ 10, 8, 6, 4, 2, 0, -2, \dots \]

The sequence is strictly decreasing and unbounded below. Hence:

\[ \lim_{n \to +\infty} a_n = -\infty \]

And so on.