Interchange of integrals and limits

[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

 

[VeeEight]

Try a sequence where convergence is not uniform

 

[g_edgar]

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

 

[symbol0]

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

 

[symbol0]

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