What is the Sum of an Infinite Geometric Progression (GP)?

A Geometric Progression (GP) is a sequence where each term is obtained by multiplying the previous term by a fixed number called the common ratio (r).

General form: a, ar, ar², ar³, ar⁴, ...

Where:

• a = first term
• r = common ratio = (any term) / (previous term)

For an infinite GP, the sequence continues indefinitely.

Examples:

• 1, 1/2, 1/4, 1/8, ... (a=1, r=1/2)
• 3, 1, 1/3, 1/9, ... (a=3, r=1/3)
• 4, −2, 1, −1/2, ... (a=4, r=−1/2)

The sum of all terms in an infinite GP is meaningful only when |r| < 1, because each successive term becomes smaller and the total approaches a finite limit.

The sum of an infinite GP with first term a and common ratio r (where |r| < 1) is:

S∞ = a / (1 − r)

Derivation: Let S = a + ar + ar² + ar³ + ar⁴ + ... (infinite sum)

Multiply both sides by r: rS = ar + ar² + ar³ + ar⁴ + ...

Subtract the second equation from the first: S − rS = a + (ar − ar) + (ar² − ar²) + ... S(1 − r) = a S = a / (1 − r)

This derivation is valid because when |r| < 1, the term rⁿ → 0 as n → ∞, so the 'tail' of the series vanishes.

When |r| ≥ 1, the terms do not shrink and the series diverges (sum is infinite or undefined).

Example 1: 1 + 1/2 + 1/4 + 1/8 + ... a = 1, r = 1/2 (|r| < 1, so convergent) S∞ = a/(1−r) = 1/(1−1/2) = 1/(1/2) = 2

Example 2: 3 + 1 + 1/3 + 1/9 + ... a = 3, r = 1/3 S∞ = 3/(1−1/3) = 3/(2/3) = 3 × 3/2 = 9/2 = 4.5

Example 3: 0.3 + 0.03 + 0.003 + 0.0003 + ... a = 0.3, r = 0.1 S∞ = 0.3/(1−0.1) = 0.3/0.9 = 1/3 = 0.333... (This shows why 0.333... = 1/3)

Example 4: Find the sum of 8 + 4 + 2 + 1 + 1/2 + ... a = 8, r = 4/8 = 1/2 S∞ = 8/(1−1/2) = 8/(1/2) = 16

Example 5: 6 − 3 + 3/2 − 3/4 + ... (r is negative) a = 6, r = −1/2 (|r| = 1/2 < 1, convergent) S∞ = 6/(1−(−1/2)) = 6/(3/2) = 6 × 2/3 = 4

The infinite GP formula S∞ = a/(1−r) is ONLY valid when |r| < 1.

Conditions:

• |r| < 1: Series converges. Sum = a/(1−r).
• |r| = 1: Series does not converge. If r = 1, all terms equal a, sum is infinite. If r = −1, terms alternate and sum oscillates.
• |r| > 1: Terms grow without bound. Sum is infinite.

Examples of non-convergent series:

• 1 + 2 + 4 + 8 + ... (r = 2): Diverges to infinity.
• 1 − 1 + 1 − 1 + ... (r = −1): Oscillates between 0 and 1, no finite sum.
• 3 + 6 + 12 + 24 + ... (r = 2): Diverges.

Intuition: For |r| < 1, each term is a fraction of the previous term, so adding infinitely many smaller and smaller numbers still gives a finite result.

Infinite GPs explain why recurring decimals equal specific fractions:

0.999... = 9/10 + 9/100 + 9/1000 + ... a = 9/10, r = 1/10 S∞ = (9/10)/(1−1/10) = (9/10)/(9/10) = 1 So 0.999... = 1 exactly.

0.333... = 3/10 + 3/100 + 3/1000 + ... a = 3/10, r = 1/10 S∞ = (3/10)/(9/10) = 3/9 = 1/3

0.142857142857... = 1/7 (can also be derived from infinite GP)

Sum of infinite GP formula for finite GP: For a finite GP with n terms: Sₙ = a(1−rⁿ)/(1−r) when r ≠ 1. As n → ∞ and |r| < 1, rⁿ → 0, so Sₙ → a/(1−r), confirming the infinite GP formula.