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Converting the integral of a sum into a sum of integrals

If I ask Mathematica to perform an indefinite integrate on a symbolic
summation (with terms of the form a_i x^i),


it does nothing.  If I try to tell it that summation and integration
commute by applying the rule:


I get a strange result equivalent to

term * range

or, in this case,

Times[Power[x,i],Subscript[a,i]] * {i,0,k}
{i a_i x^i, 0, k a_i x^i}

Any ideas as to what's going on?


1)  If you use rules to extract each of term, range, and variable and
then take Sum[Integral[...],...] the correct answer is returned.

2)  if you use a rule that changes Sum to Power, i.e.

Integrate[Sum[term_,range_],variable_] ->

you get exponents

term^(range - 1)



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