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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 created: h2013-03-2i by: hpahioi version: h41433i Privacy setting: h1i hTopici h97D99i h05A19i h00A05i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 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).