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Problem with sums differentiation


I have problems with infinite sums integration, differentiations,
multiplications when the imax is infinity or a symbol. For example

In:
a[x_,t_]=Sum[uj[t]*Cos[j Pi x],{j,1,Infinity}]; Dt[a[x,t],x]

gives me
Out: Dt[Sum[uj[t] Cos[j Pi x], {j, 1, Infinity}], x]

but not something like Sum[-uj[t]*j*Pi*Sin[j Pi x],{j,1,Infinity}]

Is there a way to move the integration (differentiation) sigh under the
sum sigh and force Mathematica to evaluate the terms?

Thank you
Dmitri Tcherniak



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