# [Maxima] integral over a finite sum with unknown upper limit

i_inbox at tn-home.de i_inbox at tn-home.de
Fri Jun 30 04:56:25 CDT 2006

```Hello,

I would like to integrate over a finite sum with not yet known upper
limit n.  Either, I get a noun limit or a noun integral that cannot be
further evaluated.

As a preparation, I've already told maxima: declare(n,integer); assume(n>0);

What I need to do is:

(%i5) integrate( sum( q[k]*s^k, k, 1, n ), s, 1, 2 );

(%o5) ('LIMIT('SUM(q[k]*s^(k+1)/(k+1),k,1,n),s,2,MINUS))
-('LIMIT('SUM(q[k]*s^(k+1)/(k+1),k,1,n),s,1,PLUS))

(in the application the summands are more complicated terms).

(%i6) ev(%,nouns);

(%o6) ('LIMIT('SUM(q[k]*s^(k+1)/(k+1),k,1,n),s,2,MINUS))
-('LIMIT('SUM(q[k]*s^(k+1)/(k+1),k,1,n),s,1,PLUS))

Why are the limits not taken?

Now, another example which fits better the application I have in mind:

(%i7) u[k](s) := s^k;

(%o7) u[k](s):=s^k

(%i8) integrate( sum( q[k]*u[k](s), k, 1, n ), s, 0, 1);

(%o8) 'INTEGRATE('SUM(q[k]*u[k](s),k,1,n),s,0,1)

(%i9) ev(%, nouns);

(%o9) 'INTEGRATE('SUM(q[k]*u[k](s),k,1,n),s,0,1)

Here, even the integration is not carried out.

But, maxima is able to integrate the summands:

(%i10) integrate( q[k]*u[k](s), s, 0, 1 );

(%o10) q[k]/(k+1)

I would like to get

sum( q[k]/(k+1), k, 1, n )

out of the input

integrate( sum( q[k]*u[k](s), k, 1, n ), s, 0, 1).

What must i tell maxima to get this right?

Your help would be very welcome.

Regards,
Tobias

```