# Measure Theory / Series of functions

## Homework Statement

I am looking for an example of a series of funtions:
$$\sum g_n$$ on $$\Re$$

such that:

$$\int_{1}^{2}\displaystyle\sum_{n=1}^{\infty}g_n(x) \, dx \neq \displaystyle\sum_{n=1}^{\infty} \, \int_{1}^{2} \, g_n(x) \, dx$$

"dx" is the Lebesque measure.

2. The attempt at a solution

I haven't attempted a solution as I'm not sure how to approach this problem. If somebody could explain this to me or link to sample problems similar to this, I would really appreciate it.

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I assume this is something that I won't be able to grasp within an hour...

Will letting $$g_n(x)=-\frac{1}{n}$$ lead anywhere?

Will letting $$g_n(x)=-\frac{1}{n}$$ lead anywhere?
No.

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I don't know, my brain is fried.

$$f_n(x)=\frac{x}{n}\, ?$$