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What is the sum to infinity of a geometric series?
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All Flashcards in Topic 1.11
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1.11.18 cards
What is the sum to infinity of a geometric series?
S∞ = u₁/(1 − r), the finite total of all (infinitely many) terms — valid only when |r| < 1.
When does a geometric series have a sum to infinity?
Only when −1 < r < 1 (|r| < 1). The terms must shrink toward 0. If |r| ≥ 1, no sum to infinity exists.
Why does an infinite series sometimes give a finite total?
When |r| < 1 each term is a fraction of the last, so the terms shrink toward 0 fast enough that the running total closes in on a fixed value.
Sum to infinity with u₁ = 12, r = 0.5?
S∞ = 12/(1 − 0.5) = 12/0.5 = 24.
How do you find r from S∞ and u₁?
Rearrange S∞ = u₁/(1 − r): 1 − r = u₁/S∞, so r = 1 − u₁/S∞.
S∞ = 36 and u₁ = 24. Find r.
1 − r = 24/36 = 2/3, so r = 1/3 (and |1/3| < 1, so it converges).
In a bouncing-ball distance problem, what's the trick?
Add the first drop, then count each rebound height TWICE (up and down): total = drop + 2 × (rebound series).
After finding S∞ in an applied question, what earns the last mark?
Interpreting it in context (e.g. the steady drug level or long-run total) and commenting on whether the model is realistic.
Topic 1.11 study notes
Full notes & explanations for Infinite geometric series (HL only)
Math AI exam skills
Paper structures, command terms & tips
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