Convergence of series.
An infinite sum can either approach a finite number (converge) or grow without bound (diverge). A few tests let you tell which — without summing the whole thing.
Geometric series
The cleanest infinite series. Start with a first term a, multiply by r each step.
a + ar + ar² + ar³ + ⋯Converges to a / (1 − r) when |r| < 1. Diverges otherwise. That's the whole story for geometric series — and it's the model every other convergence test compares against.
The n-th term test (the easy one)
If the terms don't approach zero, the sum can't converge. So if limn→∞ an ≠ 0, the series diverges.
This is a necessary condition, not sufficient: terms can shrink to zero and the series can still diverge (the harmonic series is the classic example).
Ratio test
For series with factorials or exponentials, the ratio of consecutive terms often simplifies.
L = limn→∞ |an+1 / an|If L < 1, the series converges absolutely. If L > 1 (or infinity), it diverges. If L = 1, the test is inconclusive — try another.
Worked example
Does ∑ n! / 5ⁿ converge?
an = n! / 5ⁿ |an+1 / an| = (n+1)! / 5ⁿ⁺¹ · 5ⁿ / n! = (n+1) / 5 limn→∞ (n+1) / 5 = ∞Since L = ∞ > 1, the series diverges. The factorials outpace the exponentials.
Flashcard · Unit 7
Geometric, r = 1/2. Converges to 1.
Check your understanding
Does the series ∑ n / 3ⁿ converge?