Validity & approximation
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Flip to reveal answersWhen is the extended series (1 + x)ⁿ valid?
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All 8 Flashcards — Validity & approximation
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Question
When is the extended series (1 + x)ⁿ valid?
Answer
When |x| < 1 — only then do the terms shrink so the series settles to a value.
Question
Validity of (1 + kx)ⁿ?
Answer
|kx| < 1, i.e. |x| < 1/|k|.
Question
How do you approximate a root with the series?
Answer
Choose x so the bracket equals the target number (x should be small), then evaluate the first few terms.
Question
Estimate √1.02 using (1 + x)^(1/2) ≈ 1 + ½x − ⅛x²?
Answer
x = 0.02: 1 + 0.01 − 0.00005 = 1.00995.
Question
Why does the series need |x| < 1?
Answer
Only then do the powers of x get smaller, so the later terms fade and the sum converges.
Question
Validity of (1 − 2x)⁻¹?
Answer
|2x| < 1, so |x| < ½.
Question
What x in (1 + x)^(1/2) estimates √1.04?
Answer
x = 0.04 (since 1 + x = 1.04).
Question
Does a smaller x give a better estimate?
Answer
Yes — the later (ignored) terms are even tinier, so the first few terms are more accurate.
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Full study notes for Validity & approximation
Topic 1.10 hub
Counting & binomial (HL only)
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