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empty sum∗
pahio†
2013-03-22 2:19:41
The empty sum is such a borderline case of sum where the number of the
addends is zero, i.e. the set of the addends is an empty set.
• One may think that the zeroth multiple 0a of a ring element a is the empty
sum; it can spring up by adding in the ring two multiples whose integer
coefficients are opposite numbers:
(−n)a+na = (−n+n)a = 0a
This empty sum equals the additive identity 0 of the ring, since the multiple (−n)a is defined to be
(−a)+(−a)+ . . . +(−a)
{z
}
|
n copies
• In using the sigma notation
n
X
f (i)
(1)
f (i).
(2)
i=m
one sometimes sees a case
m−1
X
i=m
It must be an empty sum, because in
m
X
f (i)
(3)
i=m
the number of addends is clearly one and therefore in (2) the number is
zero. Thus the value of (2) may be defined to be 0.
∗ hEmptySumi
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hTopici h97D99i h05A19i h00A05i
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1
Note. The sum (1) is not defined when n is less than m−1, but if one would
want that the usual rule
n
X
k
X
f (i) +
i=m
f (i) =
i=n+1
k
X
f (i)
(4)
i=m
would be true also in such a cases, then one has to define
n
X
m−1
X
f (i) = −
i=m
f (i)
(n < m−1),
i=n+1
because by (4) one could calculate
0 = −
m−1
X
i=n+1
f (i) +
m−1
X
i=n+1
f (i) =
n
X
i=m
2
f (i) +
m−1
X
i=n+1
f (i) =
m−1
X
i=m
f (i).