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Flip to reveal answersWhen does an infinite geometric series have a sum?
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All 11 Flashcards — Sum to infinity
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Question
When does an infinite geometric series have a sum?
Answer
Only when |r| < 1 — the terms shrink toward 0. Then S∞ = u₁/(1 − r).
Question
How do you turn a recurring decimal like 0.474747… into a fraction using S∞?
Answer
Write it as a GP: 0.47 + 0.0047 + … with u₁ = 0.47 and r = 0.01. Then S∞ = u₁/(1 − r) = 0.47/0.99 = 47/99. Each repeating block is the previous one × 0.01.
Question
Why must |r| < 1 for a sum to infinity?
Answer
If |r| ≥ 1 the terms don't approach zero, so the running total grows without limit — there is no finite sum.
Question
Find S∞ of 12 + 8 + 16/3 + …
Answer
r = 8/12 = ⅔, so S∞ = 12/(1 − ⅔) = 12/(⅓) = 36.
Question
Given S∞ = 25 and u₁ = 10, find r.
Answer
25 = 10/(1 − r) → 1 − r = 0.4 → r = 0.6.
Question
Given S∞ = 40 and r = 0.2, find u₁.
Answer
u₁ = S∞(1 − r) = 40 × 0.8 = 32.
Question
What's the most common S∞ mistake?
Answer
Putting r in the denominator instead of (1 − r), or using S∞ when |r| ≥ 1.
Question
How do you answer 'explain why the sum to infinity does not exist'?
Answer
State that |r| ≥ 1, so the terms do not approach zero and the total is unbounded.
Question
The partial sums approach S∞. How do you find the least n with Sₙ within a tolerance of S∞?
Answer
The gap is S∞ − Sₙ = u₁rⁿ/(1 − r). Set it below the tolerance and solve (GDC table or logs), rounding n up.
Question
If |r| ≥ 1 there is no S∞. How do you find the sum of the first 2m terms?
Answer
Use the finite sum with n = 2m: S₂ₘ = u₁(r²ᵐ − 1)/(r − 1). The power law r²ᵐ = (r²)ᵐ usually simplifies it (e.g. 3²ᵐ = 9ᵐ).
Question
How do you find the total distance a bouncing ball travels before it stops?
Answer
Total = drop + 2 × S∞ of the rebound heights, where the rebounds are a GP with first term (drop × r). For r = ½ this is 3 × the drop.
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Full study notes for Sum to infinity
Topic 1.8 hub
Infinite geometric series
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