real analysis question

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Senapboebf

Mathematics

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Explanation & Answer

||.||_1 is an integral of a function's absolute value.

Here |(f_n - f)(x)| = {0, when x from [0; 1/2] U [1/2+1/n; 1] and = n(x-1/2) for x from (1/2; 1/2+1/n).
It is a linear function at (1/2; 1/2+1/n), = 0 at 1/2 and =1 at 1/2+1/n, so the integral is the area of a right triangle,

(1/2)*1*1/n = 1/(2n). Yes, this is tends to 0 when n tends to infinity.

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We can compute the integral directly:
n*int(x-1/2) = n*(x^2/2 - x/2)  with x from 1/2 to 1/2+1/n.
This is equal to (1/2)n*[(1/2+1/n)^2 - (1/2)^2 - (1/2+1/n) + 1/2] = 
= (n/2)*[1/n + 1/n^2 + 1/4 - 1/4 -1/2 - 1/n + 1/2] = (n/2)*(1/n^2) = 1/(2n).


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