Interchange of integrals and limits

1. Aug 22, 2009

symbol0

Can anybody please show me an example of a convergent sequence of real valued functions of a real variable where the integral of the limit of the sequence is different than the limit of the integrals of functions in the sequence ?

Thank you

2. Aug 22, 2009

VeeEight

Try a sequence where convergence is not uniform

3. Aug 22, 2009

g_edgar

$$\lim _{n\rightarrow \infty } \left( \int _{0}^{1}\!{x}^{n} \left( 1-x \right) \left( n+1 \right) \left( n+2 \right) {dx} \right)$$

4. Aug 22, 2009

symbol0

So the integral of the limit is zero and the limit of the integrals is 1.
Thank you g_edgar

5. Oct 12, 2009

symbol0

If we were taking the Lebesgue integral on this sequence, we would also have the same result, right?