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From: ghgonnet@watdaisy.UUCP (Gaston H Gonnet)
Newsgroups: net.math,net.physics,net.puzzle
Subject: Re: summation in closed form (2^k/k)
Message-ID: <7512@watdaisy.UUCP>
Date: Sat, 30-Nov-85 10:00:13 EST
Article-I.D.: watdaisy.7512
Posted: Sat Nov 30 10:00:13 1985
Date-Received: Sun, 1-Dec-85 03:27:59 EST
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Xref: watmath net.math:2578 net.physics:3644 net.puzzle:1211

> 
> For k>=0, let a(k) be  2^k / k  (one over k, times the kth power of 2).
=====================
you probably mean k>0
=====================
> Can you find the sum of a(k) as k ranges from 1 to n as a closed formula
> in n?  
> 

A "closed formula" is a fuzzy concept, it depends on your set of
primitive functions.  E.g. Hn = sum(1/k,k=1..n) cannot be expressed
in terms of + - * and /, but it can be expressed in terms of psi(x).

I do not know any "closed form" for the above, if it is of any help,
it has an asymptotic expansion like:

    sum( 2^k/k, k=1..n) = 2^(n+1) / n * ( 1 + 1/n + 3/n^2 +

					  13/n^2 + 75/n^4 + O(1/n^5) )

For a more general case ( sum( z^k/k, k=1..n ) ) there is also
an asymptotic formula, see the "Handbook of Algorithms and Data
Structures", Addison Wesley, p. 210 (II.1.7))