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MathidiaMathematics Academy
Functions, iteration and numerical methods
Lesson 20 of 75
Review

Convergence and accuracy

Objective. Judge convergence and state roots to justified accuracy.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Convergence and accuracy in iteration means judging whether successive iterated values are settling towards a single fixed value, and stating the root to a precision justified by how many iterations agree. An iteration converges when the difference between consecutive terms becomes negligible, and diverges if the terms grow without bound or oscillate wildly. To state a root accurate to a given number of decimal places, continue iterating until two consecutive values round to the same figure at that precision, then confirm with a sign-change check either side. For example, if x₅=2.4142 and x₆=2.4142 to 4 d.p., the root is 2.4142 to 4 decimal places.

Worked example

An iteration gives x₄ = 1.3247, x₅ = 1.32472, x₆ = 1.32472. State the root to 4 decimal places and justify your accuracy.

An iteration gives x₄ = 1
Illustrated card showing An iteration gives x₄ = 1
  1. 1Compare x₅ and x₆: both round to 1.3247 to 4 decimal places
  2. 2Since consecutive iterations agree to this precision, the sequence has converged
  3. 3State the root using this agreed rounding
  4. 4Confirm accuracy by noting further iterations would not change the 4th decimal place

Answer: Root ≈ 1.3247 (4 d.p.)

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

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Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

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Common mistake. Pupils often stop after just one or two iterations and state that value as the final answer without checking it has stabilised, giving an inaccurate root; always continue until consecutive iterations agree to the required decimal places.

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