       Converting the integral of a sum into a sum of integrals

• To: mathgroup at smc.vnet.net
• Subject: [mg109064] Converting the integral of a sum into a sum of integrals
• From: "Kurt TeKolste" <tekolste at fastmail.net>
• Date: Mon, 12 Apr 2010 06:53:48 -0400 (EDT)
• References: <201004110831.EAA06714@smc.vnet.net>

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

Integrate[Sum[Times[Power[x,i],Subscript[a,i]],{i,0,k}],

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

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

I get a strange result equivalent to

term * range

or, in this case,

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

Any ideas as to what's going on?

(BTW:

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_] ->
Product[Integrate[term_,variable_],range_]

you get exponents

term^(range - 1)

Odd...
)

ekt

```

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