Algebra
Infinite series
Sums that go on forever — and the surprising ones that still settle on a number.
A sum of infinitely many terms. Sometimes it converges to a number, sometimes it diverges.
Try this
6
1 + ½ + ¼ + ⅛ + … → partial sum after n terms (limit 2) = 1.97
Converge or diverge?
- Converges: the partial sums settle toward a single number (the sum). E.g. 1 + ½ + ¼ + … → 2.
- Diverges: the partial sums grow without bound, or never settle. E.g. 1 + 1 + 1 + … or the harmonic series 1 + ½ + ⅓ + …
- A necessary check: if the terms don't shrink to 0, the series cannot converge (but shrinking terms alone isn't enough — see the harmonic series).
The harmonic series 1 + ½ + ⅓ + ¼ + … diverges, but excruciatingly slowly — you need over 10⁴³ terms just to pass a total of 100.
Your turn
Does 2 + 2/3 + 2/9 + 2/27 + … converge? To what?
1 + 1/2 + 1/4 + 1/8 + ... = 2. But 1 + 1/2 + 1/3 + 1/4 + ... = ∞ (the harmonic series diverges).