The terms must shrink: The infinite series only settles to a value when each term is smaller than the last — that needs |x| < 1.
For (1 + kx)ⁿ the inside term must be small: |kx| < 1, i.e. |x| < 1/|k|.
IB-style question — range of validity
State the values of x for which the expansion of (1 + 3x)⁻² is valid.
Step by step
- The inside term is 3x; it must satisfy |inside| < 1.
- Divide by 3.
Final answer
|x| < ⅓ (that is, −⅓ < x < ⅓).
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Pick a small x, plug it in: Choose x so the bracket equals the number you want, then put that small x into the first few terms for a quick decimal estimate.
IB-style question — estimate a square root
Given that (1 + x)1/2 ≈ 1 + ½x − ⅛x², use x = 0.02 to estimate √1.02.
Step by step
- Check: 1 + x = 1.02 gives x = 0.02, which is small (valid).
- Substitute into the three terms.
- Evaluate.
Final answer
√1.02 ≈ 1.00995.