Nice2

LevonNurbekian

Here is another nice problem.
Give an example of functional sequence of integrable functions ${f_{n}}(x)$ defined on the segment $[0,1]$ ,such that $\lim_{n{\rightarrow}{\infty}}{f_{n}(x)}=+\infty,\forall x\in{[0,1]}$ ,but $\lim_{n{\rightarrow}{\infty}}{\int_{0}^{1}{f_{n}(x)dx}}=-\infty$ .

**Posted:** Wed Nov 17, 2004 12:05 pm

Myth

Please, don't blame me, but I have nothing to say except it is too obvious.

**Posted:** Wed Nov 17, 2004 12:23 pm

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Myth is out of here

Kent Merryfield

I get the feeling that "obvious" to Myth isn't quite the same as obvious to the average reader of this board. Hence, it's worth writing out an example. I'll even make the functions continuous. For $n\ge2$ let $f_n(x)$ be a piecewise continuous linear function whose graph has the following vertices: $(0,n), \left(\frac1n,n-n^3\right), \left(\frac2n,n\right), (1,n).$ This tends pointwise to $\infty$ - that is, for all $x\in[0,1], \lim_{n\to\infty}f(x)=\infty.$ This is because for $x>0$ , the sequence eventually takes the value $n,$ and it is $n$ for $x=0.$ Howver the integral is the integral of $n$ minus the area of a triangle of base $\frac2n$ and height $n^3.$ That is, $\int_0^1f_n(x)dx= n-n^2$ and that tends to $-\infty.$

**Posted:** Wed Nov 17, 2004 2:37 pm

Myth

I meant the same example. It is very standart and easy.

**Posted:** Wed Nov 17, 2004 9:50 pm

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Myth is out of here