Mathematics · Advanced university
Uniform error of a function sequence
A bound uniform in x proves more than pointwise convergence at a chosen x.
Method and conditions
A bound uniform in x proves more than pointwise convergence at a chosen x.
||fₙ||∞ = sup |fₙ(x)|
Worked examples
For f_n(x) = 25x/n on [0,1], calculate sup over x of |f_n(x)| when n = 72. Round the final result to 4 decimal places if needed.
The function is nonnegative and increasing in x; its maximum occurs at x = 1 and equals 25/72.
0.3472
For f_n(x) = 23x/n on [0,1], calculate sup over x of |f_n(x)| when n = 73. Round the final result to 4 decimal places if needed.
The function is nonnegative and increasing in x; its maximum occurs at x = 1 and equals 23/73.
0.3151
Try it
For f_n(x) = 21x/n on [0,1], calculate sup over x of |f_n(x)| when n = 6. Round the final result to 4 decimal places if needed.
For f_n(x) = 19x/n on [0,1], calculate sup over x of |f_n(x)| when n = 9. Round the final result to 4 decimal places if needed.
Check your work
Answers and marking notes
- 3.5
The function is nonnegative and increasing in x; its maximum occurs at x = 1 and equals 21/6.
- 2.1111
The function is nonnegative and increasing in x; its maximum occurs at x = 1 and equals 19/9.
Further reading and review references
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