Terms and limits of a geometric sequence

Direction of Variation

The behaviour of a geometric sequence (u_n) depends on the sign of u_0 and the value of the common ratio q:

Condition	Behaviour
u_0 > 0 and q > 1	Increasing sequence, tends to +∞
u_0 > 0 and 0 < q < 1	Decreasing sequence, tends to 0
u_0 > 0 and q < 0	Alternating sequence (changes sign with each term)
q = 1	Constant sequence

Example of convergence

Let u_0 = 100 and q = 0.5. The terms are 100, 50, 25, 12.5, 6.25, … The sequence decreases and gets ever closer to 0 without ever reaching it. We say that it converges to 0.

Example of divergence

Let u_0 = 1 and q = 2. The terms are 1, 2, 4, 8, 16, 32, … The sequence increases indefinitely: it diverges to +∞.

Rule to remember about limits

If -1 < q < 1 (i.e. |q| < 1), then q^n tends towards 0 as n becomes large, and the geometric sequence converges to 0 (assuming u_0 is not equal to 0).

If q > 1, q^n tends towards +∞: the sequence diverges.

If q ≤ -1 or q = 1 with u_0 not equal to 0, the behaviour is either constant or oscillating without a limit.

Common pitfall

A negative ratio such as q = -2 does not mean that the sequence is decreasing: it alternates between positive and negative values, with increasing absolute values. Do not confuse a ‘negative ratio’ with a ‘decreasing sequence’.