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From: leimkuhl@uiucdcsp.CS.UIUC.EDU
Newsgroups: net.math
Subject: Re: summation in closed form
Message-ID: <9600034@uiucdcsp>
Date: Sun, 1-Dec-85 23:12:00 EST
Article-I.D.: uiucdcsp.9600034
Posted: Sun Dec  1 23:12:00 1985
Date-Received: Tue, 3-Dec-85 07:57:58 EST
References: <2580@sjuvax.UUCP>
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Nf-From: uiucdcsp.CS.UIUC.EDU!leimkuhl    Dec  1 22:12:00 1985




Well the given problem can be looked at as an integration problem:

   Sum{x^k/k}   =  Sum{   Int{ s^(k-1)} - 1/k}
  k=1,n		 k=1,n   1<s<=x

		=  Int{   Sum{ s^k}} - Sum{1/k}
		 1<s<=x  k=0,n-1      k=1,n

		=  Int{(s^n - 1)/(s-1)}  - Sum{1/k}
		 1<s<=x			  k=1,n


Where by Int{f(s)}, we mean lim{    Int{f(s)}}.  Since f is continuous
	a<s<=b		   u->a+  u<=s<=b
and bounded in our case, this makes sense. 

I let Macsyma have a crack at the integral, but it proved too tough.  For this
reason, I doubt the existence of a simple closed form.  There is always
the problem of determining what functions are in the space of acceptable
functions (indeed, in one sense the given sum is already in closed form, 
for it has a finite number of closed terms.)  

Does anyone have a good book of integrals handy?

-Ben Leimkuhler