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From: bhuber@sjuvax.UUCP (B. Huber)
Newsgroups: net.math,net.physics,net.puzzle
Subject: summation in closed form
Message-ID: <2580@sjuvax.UUCP>
Date: Wed, 27-Nov-85 10:34:17 EST
Article-I.D.: sjuvax.2580
Posted: Wed Nov 27 10:34:17 1985
Date-Received: Sat, 30-Nov-85 00:35:27 EST
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Organization: St. Joseph's University, Phila. PA.
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Xref: watmath net.math:2571 net.physics:3641 net.puzzle:1208


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

I suspect not, but cannot yet find a way to show that it is impossible.

The answer is of some interest.  It would yield a closed formula for the
sums of reciprocals of entries in Pascal's triangle (summed over individual
rows).  Thus it would give a formula for the resistance through a circuit
shaped like an n-cube (a problem which appeared in net.physics two months
ago).

A nice little mathematical puzzle is to find the exact relationship between
the two sums which I have mentioned.  In particular, can you find it using
only elementary techniques (algebraic, not transcendental; formal power
series, etc., not allowed)?