Extended binomial theorem
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Flip to reveal answersWhen does the binomial expansion become an infinite series?
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All 8 Flashcards — Extended binomial theorem
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Question
When does the binomial expansion become an infinite series?
Answer
When n is negative or a fraction (not a positive whole number) — the terms never stop.
Question
Extended binomial formula for (1 + x)ⁿ?
Answer
1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … , valid for |x| < 1.
Question
How do you build each successive coefficient?
Answer
Multiply by the next falling factor of n on top and divide by the next factorial: n, then n(n−1)/2!, then n(n−1)(n−2)/3!.
Question
How do you expand (1 + kx)ⁿ?
Answer
Replace every x in the formula with kx, then simplify — remember to square and cube the k.
Question
First three terms of (1 + x)⁻¹?
Answer
1 − x + x².
Question
First three terms of (1 − 3x)⁻²?
Answer
1 + 6x + 27x².
Question
Coefficient of x in (1 + x)ⁿ?
Answer
n.
Question
Coefficient of x² in (1 + x)ⁿ?
Answer
n(n − 1)/2.
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Full study notes for Extended binomial theorem
Topic 1.10 hub
Counting & binomial (HL only)
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