MathIsland YouTubeYouTube

PROOF 38

Prove.

If the common ratio is less than 1, the infinite geometric series converges

On the real numbers ℝ.

What you may use

A10<a, 0<b ⟹ some n with na > b
Archimedean property
A2(1+h)ⁿ ≥ 1 + nh (h>0)
Bernoulli's inequality
A30<c ⟹ (a<b ⟺ ac<bc)
Order axiom (positive multiplication)
A4a ≤ b ≤ c, one strict ⟹ a<c
Transitivity of order
A50 < 1, a < 1 + a
Order axiom (1 is positive)
A6a < b ⟹ some h>0 with b = a+h
Existence of the difference
A70 < a < 1 ⟹ 1 < 1/a
Order of reciprocals
A8a < a never holds
Order axiom (irreflexivity)
A9a ≠ 0 ⟹ a·(1/a) = 1
Multiplicative inverse
A100 ≤ |a|, a ≤ |a|
Meaning of absolute value
A11|a| = 0 ⟺ a = 0
Absolute value and 0
A12|ab| = |a| |b|
Absolute value and products
A13P(1), P(k)→P(k+1)⟹∀nP(n)
Induction axiom
A14Sₙ = a(1−rⁿ)/(1−r)
Sum of a geometric sequence (Proof 37, r ≠ 1)
A15lim c·xₙ = c·lim xₙ
Constant multiple of a limit
A16lim(c − xₙ) = c − lim xₙ
Limit and difference with a constant
A17xₙ → L ⟹ xₙ₊₁ − xₙ → 0
Adjacent differences of a convergent sequence
A18a × 0 = 0
Multiplying by 0 gives 0
A19a − 0 = a
Subtracting 0 changes nothing
A20(ab)ⁿ = aⁿbⁿ, 1ⁿ = 1
Laws of exponents
A21(ab)c = a(bc)
Associativity of multiplication
A22ab = ba
Commutativity of multiplication
A23(ab)/c = (a/c)b
Placement of division
A24a(b − c) = ab − ac
Distributive law (subtraction form)
A25a × 1 = a
Multiplicative identity
A26(a + b) − a = b
Undoing a subtraction
Ea = b ⟹ same operation on both sides
Properties of equality

Nothing outside this list may be cited.

Definition D1

The value of the infinite geometric series a + ar + ar² + … is defined as the limit of the partial-sum sequence (Sₙ).

Proof.

Related labs

Limit lab →Sequence lab →