Concrete mathematics : a foundation for computer science

2
Sums
SUMS ARE EVERYWHERE in mathematics, so we need basic tools to handle
them. This chapter develops the notation and general techniques that make
summation user-friendly.
2.1 NOTATION
In Chapter 1 we encountered the sum of the first n integers, which
wewroteoutas1+2+3+...+(n-1)fn. The‘...‘insuchformulastells
us to complete the pattern established by the surrounding terms. Of course
we have to watch out for sums like 1 + 7 + . . . + 41.7, which are meaningless
without a mitigating context. On the other hand, the inclusion of terms like
3 and (n - 1) was a bit of overkill; the pattern would presumably have been
clear if we had written simply 1 + 2 + . . . + n. Sometimes we might even be
so bold as to write just 1 f.. . + n.
We’ll be working with sums of the general form
al + a2 + ... + a,, (2.1)
where each ok is a number that has been defined somehow. This notation has
the advantage that we can “see” the whole sum, almost as if it were written
out in full, if we have a good enough imagination.
A term is how long Each element ok of a sum is called a term. The terms are often specified
this course lasts. implicitly as formulas that follow a readily perceived pattern, and in such cases
we must sometimes write them in an expanded form so that the meaning is
clear. For example, if
1 +2+ . . . +2+'
is supposed to denote a sum of n terms, not of 2”-‘, we should write it more
explicitly as
2O + 2' +. . . + 2n-'.
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