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From: eirik@tekchips.UUCP (Eirik Fuller)
Newsgroups: net.math
Subject: Re: series sum
Message-ID: <17@tekchips.UUCP>
Date: Sun, 26-Jan-86 23:15:30 EST
Article-I.D.: tekchips.17
Posted: Sun Jan 26 23:15:30 1986
Date-Received: Thu, 30-Jan-86 04:39:36 EST
References: <2265@utcsstat.uucp>
Reply-To: eirik@tekchips.UUCP (Eirik Fuller)
Organization: Tektronix, Beaverton OR
Lines: 63

In article <2265@utcsstat.uucp> anthony@utcsstat.uucp
	(Anthony Ayiomamitis) writes:
>
>	Can anyone deduce the sum of the following series after n terms?
>
>	  1 + 3 + 6 + 10 + 15 + 21 + .....
>
>-- 
>
>       	{allegra,ihnp4,linus,decvax}!utzoo!utcsstat!anthony
>        {ihnp4|decvax|utzoo|utcsrgv}!utcs!utzoo!utcsstat!anthony

Well, that depends on what the rest of the terms are ...  8^).

The Nth partial sum of this series is S(N) = N(N+1)(N+2)/6.

Proof: True for N=1 (check it out).

S(N) - S(N-1) = N(N+1)[(N+2)-(N-1)]/6 = N(N+1)/2, the Nth term in
the sum.

By induction, the formula holds.


Oh yeah, to convince yourself that the Nth term is N(N+1)/2 :

The most obvious pattern I detect in the terms is that the Nth term
differs from the (N-1)th term by N, so the Nth term is the sum of
the first N integers. Using the same idea as above, N=1 checks, and

[N(N+1)/2] - [(N-1)N/2] = N[(N+1)-(N-1)]/2 = N.


Sorry for the length of this (well, maybe ... :-), but I have two
more observations.

Long ago I worked out a formula for summing the first N integers,
each raised to the Kth power (sum from I = 1 to N of [I to the K], in
clumsy ASCII notation :-), where K is a positive integer.  It
involves solving a linear set of equations with Pascal's triangle in
a triangular matrix; beautiful tidbit, or so I thought when I
discovered it.  With even the slightest bit of encouragement (DON'T
POST; mail only please) I could be pursuaded to post it, but I fear
it might be common knowledge.  Anyway, this formula is what I used to
find the answer to this problem (I had to re-derive it though).  

The other observation. My crack about knowing the other terms in the
series was half serious. (Seriesly ... :-). Given any series
specified by a finite number of terms, it is trivially easy to
generate a closed form equation for an infinite number of other
series beginning with the same finite set of initial terms. So which
of all possible series is meant in any one case? It depends on being
able to recognize the pattern. Of course, I am presumptuous enough
to assume that I guessed the intent of this one. My formula works
for all of the given terms anyway 8^).


One more brief (?) closing comment: my original excursion into this
was triggered by elementary integration in beginner calculus. These
formulas can be used (without the fundamental theorem of calculus)
to integrate polynomials using the definition of integration as a
limiting sum of rectangles. Any calculus text I've seen has these
formulas for K=1,2, and 3, generally with little or no proof.