Sequences and Series (A2 Unit 3: Pure Mathematics B) — WJEC A-Level Mathematics
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Sequences and Series (A2 Unit 3: Pure Mathematics B) explained
Unlike the finite expansion for positive integers, the binomial expansion for negative or fractional n produces an infinite series that converges if and only if the absolute value of the variable term is strictly less than 1: |X| < 1.
Read the full explanation
For the standard form (1 + X)ⁿ, convergence requires |X| < 1, which corresponds to the open interval −1 < X < 1. When expanding (a + bx)ⁿ = aⁿ(1 + (b/a)x)ⁿ, the variable term is X = (b/a)x, so the condition for validity is |(b/a)x| < 1, which rearranges to |x| < |a/b|, or −|a/b| < x < |a/b|. If a value of x outside this open interval is substituted, the series diverges and generates meaningless or infinite values; valid approximations require choosing substitutions where |bx/a| is comfortably small.
Your focus
- State the convergence condition |X| < 1 for infinite binomial series.
- Determine the range of validity for expansions of the form (a + bx)ⁿ.
- Explain why substituting values outside the range of validity produces invalid results.
Sequences and Series (A2 Unit 3: Pure Mathematics B) exam tips
Marking Points
- stating the convergence condition |X| < 1 for the scaled variable term
- substituting X = (b/a)x to form the inequality |(b/a)x| < 1
- stating the exact range of validity as |x| < |a/b| or −|a/b| < x < |a/b|
Examiner Tips
- 💡Identify the term that plays the role of X in (1 + X)ⁿ and write |X| < 1 immediately.
- 💡State the validity condition using either modulus notation |x| < k or open interval −k < x < k.
Common Mistakes
- stating the validity condition as |x| < 1 instead of scaling by the coefficient ratio |a/b|
- using non-strict inequalities (≤) instead of strict inequalities (<) for convergence