1045 11.1 Sequences and Series (b) a 3 i=1 16xi - 22 = 16x1 - 22 + 16x2 - 22 + 16x3 - 22 Let i = 1, 2, and 3, respectively. = 16 # 2 - 22 + 16 # 4 - 22 + 16 # 6 - 22 Substitute the given values for x1, x2, and x3. = 10 + 22 + 34 Multiply and subtract inside the parentheses. = 66 Add. (c) a 4 i=1 ƒ1xi2Δx = ƒ1x12Δx + ƒ1x22Δx + ƒ1x32Δx + ƒ1x42Δx Let i = 1, 2, 3, and 4. = x1 2Δx + x 2 2Δx + x 3 2Δx + x 4 2Δx ƒ1x2 = x2 = 02122 + 22122 + 42122 + 62122 Substitute the given values for x1, x2, x3, and x4, with Δx = 2. = 0 + 8 + 32 + 72 Simplify. = 112 Add. S Now Try Exercises 59, 61, and 71. Summation Properties If a1, a2, a3, c , an and b1, b2, b3, c , bn are two sequences, and c is a constant, then for every positive integer n, the following hold true. (a) a n i=1 c =nc (b) a n i=1 cai =c a n i=1 ai (c) a n i=1 1 ai +bi 2 = a n i=1 ai + a n i=1 bi (d) a n i=1 1 ai −bi 2 = a n i=1 ai − a n i=1 bi Proof To prove Property (a), expand the series. an i=1 c = c + c + c + c + g + c n terms of c = nc Proof Property (c) also can be proved by first expanding the series. an i=1 1ai + bi2 = 1a1 + b12 + 1a2 + b22 + g + 1an + bn2 = 1a1 + a2 + g + an2 + 1b1 + b2 + g + bn2 Commutative and associative properties = a n i=1 ai + a n i=1 bi Proofs of the other two properties are similar. Summation Properties and Rules These provide useful shortcuts. LOOKING AHEAD TO CALCULUS Summation notation is used in calculus to describe the area under a curve, to describe the volume of a figure rotated about an axis, and in many other applications, as well as in the definition of integral. In the definition of the definite integral, g is replaced with an elongated S: L b a ƒ1x2 dx = lim nS∞ a n i=1 ƒ1xi2 Δxi. In some cases, the definite integral can be interpreted as the sum of the areas of rectangles.
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