Sequences and series

A series adds the terms of a sequence: a₁ + a₂ + a₃ + …. Can infinitely many numbers add up to something finite? Sometimes.

Geometric series

Each term is the previous one times a ratio r: a + ar + ar² + …. When r is between −1 and 1, the terms shrink fast enough and the sum converges to

a ÷ (1 − r)

So 1/2 + 1/4 + 1/8 + … = (1/2) ÷ (1 − 1/2) = 1. When r is 1 or more in size, the sum diverges: it never settles.

Shrinking terms are not enough

The harmonic series 1 + 1/2 + 1/3 + 1/4 + … diverges, although its terms shrink to 0. It grows very slowly: the first million terms add up to only about 14.4, but it passes every number eventually.

Taylor series

Many functions are infinite sums of powers. The most famous is

eˣ = 1 + x + x²/2! + x³/3! + …

At x = 1 the terms are 1/k!, which shrink very fast. Eight terms already give e to four decimals.

from math import e, factorial

total = 0
for k in range(1, 11):
    total += 1 / 2 ** k
print("1/2 + 1/4 + ... (10 terms) =", round(total, 6))

approx = 0
for k in range(8):
    approx += 1 / factorial(k)
    print(k + 1, "terms of the series for e:", round(approx, 6))
print("math.e =", round(e, 6))

Calculators and the math library compute sin, cos and exp from series like these.

Common mistakes

• Using a ÷ (1 − r) when r is not between −1 and 1. For r = 2 it gives −a, which is nonsense for a sum of growing positive numbers.

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