Series validity
Objective. State intervals and control approximation error.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A Maclaurin or other power series does not necessarily converge to the function for every x; there is typically an interval of validity outside of which the series diverges or no longer represents the function accurately. For series like ln(1+x) or (1+x)ⁿ for non-integer n, the interval is often −1 < x ≤ 1 or −1 < x < 1, found using the ratio test or by knowing where the original function is defined. Stating the interval of validity alongside a series is essential, since using the expansion outside this range gives meaningless or wildly inaccurate results.
Worked example
State the series expansion of ln(1+x) up to the term in x³, and give its interval of validity.
- 1Differentiating repeatedly: f(0)=0, f'(0)=1, f''(0)=−1, f'''(0)=2
- 2Build terms: 0 + x + (−1)x²/2! + 2x³/3! = x − x²/2 + x³/3
- 3This matches the known expansion ln(1+x) = x − x²/2 + x³/3 − ...
- 4The series converges for −1 < x ≤ 1, since ln(1+x) is undefined for x ≤ −1 and the series diverges for x>1
Answer: ln(1+x) ≈ x − x²/2 + x³/3, valid for −1 < x ≤ 1
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Start practisingCommon mistake. Students assume a series is valid for all real x just because it was derived from repeated differentiation. Always state and justify the interval of validity, as many series (e.g. ln(1+x)) only converge for restricted x.
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