AI generatedAlgebraGCSE Higher

Interpreting convergence of iteration

Use successive values to justify a requested accuracy.

  • Use the previous answer as the next input. Keep unrounded calculator values.
  • Round only the values you report.
  • Convergence means successive values approach a limit; a few matching digits alone are not a proof.

Example 1

Describe the convergence and estimate the positive root being approached.

Successive iterates (6 d.p.)
61.414214
71.414214
81.414214
91.414214
101.414214
At a limit, the next value equals the current value.Subtract L from both sides.Multiply both sides by L.The iterates are positive, so use the positive root.
The values settle towards ≈ 1.414.

Example 2

Describe the convergence and estimate the positive root being approached.

Successive iterates (6 d.p.)
61.732051
71.732051
81.732051
91.732051
101.732051
At a limit, the next value equals the current value.Subtract L from both sides.Multiply both sides by L.The iterates are positive, so use the positive root.
The values settle towards ≈ 1.732.

Your turn

Question 1

Describe the convergence and estimate the positive root being approached.

Successive iterates (6 d.p.)
62.000000
72.000000
82.000000
92.000000
102.000000
Check answer
At a limit, the next value equals the current value.Subtract L from both sides.Multiply both sides by L.The iterates are positive, so use the positive root.
The values settle towards ≈ 2.000.
Question 2

Describe the convergence and estimate the positive root being approached.

Successive iterates (6 d.p.)
62.236068
72.236068
82.236068
92.236068
102.236068
Check answer
At a limit, the next value equals the current value.Subtract L from both sides.Multiply both sides by L.The iterates are positive, so use the positive root.
The values settle towards ≈ 2.236.
Question 3

Describe the convergence and estimate the positive root being approached.

Successive iterates (6 d.p.)
62.449490
72.449490
82.449490
92.449490
102.449490
Check answer
At a limit, the next value equals the current value.Subtract L from both sides.Multiply both sides by L.The iterates are positive, so use the positive root.
The values settle towards ≈ 2.449.
Question 4

Describe the convergence and estimate the positive root being approached.

Successive iterates (6 d.p.)
62.645751
72.645751
82.645751
92.645751
102.645751
Check answer
At a limit, the next value equals the current value.Subtract L from both sides.Multiply both sides by L.The iterates are positive, so use the positive root.
The values settle towards ≈ 2.646.
Question 5

Describe the convergence and estimate the positive root being approached.

Successive iterates (6 d.p.)
62.828427
72.828427
82.828427
92.828427
102.828427
Check answer
At a limit, the next value equals the current value.Subtract L from both sides.Multiply both sides by L.The iterates are positive, so use the positive root.
The values settle towards ≈ 2.828.
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