Question

Explain how to tell if a geometric series is convergent or divergent.
Include an example of a convergent series and an example of a divergent series in your explanation.

Answer 1

A geometric series has the form
S = sum_{n=0}^\infty a r^n
(or equivalently sum from n=1 with ar^{n-1}). To decide convergence use the common ratio r:

- If |r| < 1 the series converges and its sum is
S = a/(1 − r).
(This follows from the finite-sum formula S_N = a(1 − r^{N+1})/(1 − r) and letting N → ∞ so r^{N+1} → 0.)

- If |r| ≥ 1 the series diverges (except the trivial case a = 0, which gives the zero series).

Examples
- Convergent: sum_{n=0}^\infty 3(1/2)^n. Here a = 3, r = 1/2 and |r| < 1, so the sum = 3/(1 − 1/2) = 6.
- Divergent: sum_{n=0}^\infty 2·2^n. Here r = 2 and |r| > 1, so terms do not go to 0 and partial sums grow without bound — the series diverges.
(Another common divergent example: sum_{n=0}^\infty (-1)^n, r = −1, whose partial sums oscillate and do not have a limit.)