Calculus 2 — Integrals & Series

Series — infinite sums with finite answers

continues from lesson 2 — values defined earlier in the course stay live here

Add 1 + ½ + ¼ + ⅛ + ⋯ forever and you do NOT get infinity — each term covers half the remaining distance to 2. A geometric series with ratio |r| < 1 converges to 1/(1−r):

S_{20} := Sum("0.5^i", "i", 0, 20)

2.0000

21 terms in — already at 1.9999995

✓ pass

\mathrm{abs}\left(S_{20} - 2\right) < 0.00001

converging to exactly 2

But not every shrinking-term series converges! The harmonic series 1 + ½ + ⅓ + ¼ + ⋯ grows without bound — glacially. After 1000 terms it has only reached ~7.49:

H_{1000} := Sum("1/i", "i", 1, 1000)

7.4855

still climbing — it never stops