By L. Auslander, R. Tolimieri

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1 k 10 k prime The advantage of this notation—called the standard form of the sigma-notation—is that the summation index is no longer restricted to a sequence of consecutive integers. Example. Consider the following manipulation: 2k+2 = −2 k n−3 2k+2 = 0 k+2 n−1 2 j = 2n − 1. 7) j n−1 After adding 2 to each term in the inequalities, we have simply replaced k + 2 with j, obtaining the sum of a geometric progression which is evaluated explicitly. With this notation the change of summation index is unproblematic.

J=1 I suggest that you adopt a summation symbol (one of i, j, k, l, m, n, see Sect. 2) and stick to it, unless there is a good reason to change it. 5). In an unrestricted sum, range information may be omitted altogether: k n = 2n n k 0. ) The summation range may also be specified by inequalities placed below the summation symbols: ak a j,k . ak 1 k n k 1 1 j,k N Further conditions may be added to alter the range of summation: ak = a−2 + a−1 + a1 + a2 0<|k| 2 1 Jean Baptiste Joseph Fourier (French: 1768–1830).

An infinite sequence of binomials with integer coefficients with unbounded coefficients with increasing degree whose leading term alternates in sign. 2 Describing Expressions We expand our dictionary with terms describing broad attributes of expressions. An expression involving numbers, the four arithmetical operations, and raising to an integer or fractional power (extraction of roots), is called an arithmetical expression. The value of an arithmetical expression is a number. A combination of rational numbers and square roots of rational numbers is called a quadratic irrational or a quadratic surd.