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What condition must hold for S∞ to exist?
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All Flashcards in Topic 1.3
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1.3.48 cards
What condition must hold for S∞ to exist?
|r| < 1 — the terms must be getting smaller toward zero.
Think: what happens to terms if r = 2 vs r = 0.5?
Write the Sum to Infinity formula.
S∞ = u₁ ÷ (1 − r). Only valid when |r| < 1.
The denominator is (1 − r), not r.
Does S∞ exist for: 3 + 6 + 12 + 24 + ... ?
No. r = 6 ÷ 3 = 2. |r| = 2 ≥ 1, so S∞ does not exist.
Find r first, then check |r|.
Does S∞ exist for: 10 + 5 + 2.5 + ... ? If yes, find it.
r = 0.5. |r| = 0.5 < 1 ✓. S∞ = 10 ÷ (1 − 0.5) = 20.
Check |r| < 1 first, then apply the formula.
S∞ = 30 and r = 0.4. Find u₁.
u₁ = S∞ × (1 − r) = 30 × (1 − 0.4) = 30 × 0.6 = 18.
Rearrange: multiply both sides by (1 − r).
u₁ = 12 and S∞ = 20. Find r.
1 − r = u₁ ÷ S∞ = 12 ÷ 20 = 0.6, so r = 0.4.
Sub into S∞ = u₁ ÷ (1 − r) and isolate r.
r = −0.6. Does S∞ exist? Explain.
Yes. |r| = |−0.6| = 0.6 < 1 ✓. Negative r is fine — |r| strips the sign.
|r| means absolute value. Strip the minus.
Exam rule: what must you write before calculating S∞?
State: |r| < 1 ✓. IB mark schemes award this step — you earn the method mark even if the final answer is wrong.
Never skip the check. It is worth marks on its own.
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