Suppose f_n(x)=x^n-x^(2n), x in [0,1]. Dose the sequence of functions f_n converge uniformly?

Kale Sampson

Kale Sampson

Answered question

2022-11-14

Suppose f n ( x ) = x n x 2 n . Dose the sequence of functions { f n } converge uniformly?

Answer & Explanation

Leo Robinson

Leo Robinson

Beginner2022-11-15Added 14 answers

The sequence ( f n ) is pointwise convergent to the zero function f on [0,1] and by derivative we have
f n ( x ) = x n 1 ( n 2 n x n ) = 0 x = 0 or x = 1 2 n
hence we find
| | f n f | | = sup x [ 0 , 1 ] | f n ( x ) f ( x ) | = f n ( 1 2 n ) = 1 4 0
so ( f n ) does not converges uniformly to f on [ 0 , 1 ]

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