Limits of Geometric Sequences as n Grows

The limits of geometric sequences as n grows is a straightforward classification: for a geometric sequence a_n = a * r^n, the limit as n approaches infinity is zero when |r| < 1, equals the initial term a when r = 1, oscillates between a and -a with no limit when r = -1, and diverges (does not exist) when |r| > 1. The initial term a only affects the scale, not the limit behaviour, unless a = 0, in which case every term is zero and the limit is zero regardless of r. This classification tells you exactly when a repeated-ratio sequence approaches zero, a finite value, oscillation, or infinity.

Take a = 3 and r = 0.5. The sequence terms are: n=0 gives 3, n=1 gives 1.5, n=2 gives 0.75, n=3 gives 0.375, n=4 gives 0.1875, n=5 gives 0.09375. Each step multiplies by 0.5, halving the previous term. As n grows, the terms shrink toward zero. The limit is 0. This is the case for any common ratio r with an absolute value less than 1, regardless of whether r is positive or negative. For r = 0.5, the approach is monotonic and smooth. For r = -0.5, the terms alternate sign but still shrink in absolute value toward zero. The calculation is immediate: multiply the initial term by the common ratio raised to increasingly large powers. When you need to verify a specific value, a geometric sequence calculator handles the arithmetic, but the limit itself is determined by the magnitude of r alone.

The calculation rule comes from Stewart Calculus §11.1: for a geometric sequence a_n = a * r^n, the limit depends only on r. When |r| < 1, r^n approaches 0, so a_n approaches 0. When r = 1, r^n = 1 for every n, so a_n = a. When r = -1, r^n alternates between 1 and -1, so the sequence bounces between a and -a without settling on a single value; the limit does not exist. When |r| > 1, the absolute value of r^n grows without bound, so the sequence diverges to infinity (if r > 1) or oscillates with unbounded magnitude (if r < -1). In both cases the limit does not exist as a finite number.

A specific failure case: r = -2 with a = 1. The terms are n=0: 1, n=1: -2, n=2: 4, n=3: -8, n=4: 16, n=5: -32. The absolute values grow without bound, but the sign alternates every term. The limit does not exist. It is not an infinite limit because the sequence does not approach a single direction; it jumps between positive and negative values that are increasingly far apart. This is a common confusion pair: students mistake unbounded growth for an infinite limit, but infinite limits require the function to approach a single signed infinity, not to alternate.

The reason r dictates the limit is that multiplication by a fixed ratio repeatedly either shrinks, stabilises, or amplifies the term. This is the core insight of the limits of geometric sequences as n grows explained in a single sentence: the repeated ratio acts as a scaling factor that either compresses the sequence toward zero, holds it steady, or blows it up. The initial term a only scales the starting point; if a = 0, every term is 0 and the limit is 0 for any r. If a is nonzero, the limit behaviour is entirely controlled by the magnitude and sign of r.

This classification provides a focused bridge between geometric patterns and calculus. The intuition that repeated multiplication by a number less than 1 in absolute value drives terms to zero is the same intuition that underlies convergence of geometric series, compound interest decay, and radioactive half-life models. The limit exists as a finite number only in the first two cases (|r| < 1 and r = 1). For |r| > 1, the sequence diverges, and for r = -1, it oscillates without approaching any single value. This is the complete answer to the question: when does a repeated-ratio sequence approach zero, a finite value, oscillation, or infinity?

The single thing that most often goes wrong is confusing the limit of a geometric sequence with the sum of a geometric series. A sequence limit asks what happens to the individual terms as n grows. A series sum asks what happens when you add all the terms together. For the sequence a_n = a * r^n with |r| < 1, the terms approach zero, but the sum of the infinite series a + ar + ar^2 + ... converges to a/(1-r). These are different questions with different answers. The sequence limit is about the terms themselves; the series sum is about their accumulation. The classification applies only to the sequence limit, not to the series sum.

Another failure mode: assuming that because a sequence oscillates, it cannot have a limit. That is correct for oscillation between two fixed values like a and -a when r = -1, but oscillation that decays in amplitude, such as r = -0.5, still has a limit of 0 because the oscillation shrinks to zero. The key is whether the oscillation dampens or persists. Persisting oscillation with constant amplitude means no limit. Decaying oscillation means the limit is 0.