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Welcome back! July 5 7.1–7.6. Sigma notation, supremum, infimum, and the definition of the integral Warm-up: sums Compute 1. 4X i=2 (2i + 1) 2. 4X i=2 2i + 1 3. 4X j=2 (2i + 1) Write these sums with notation 1. 15 + 25 + 35 + 45 + . . .+ 1005 2. 2 42 + 2 52 + 2 62 + 2 72 + . . .+ 2 N2 3. cos 0 cos 1 + cos 2 cos 3 + . . .± cos(N + 1) 4. 1 0! + 1 2! + 1 4! + 1 6! + . . .+ 1 (2N)! 5. 1 1! 1 3! + 1 5! 1 7! + . . .+ 1 81! 6. 2×3 4! + 3×4 5! + 4×5 6! + . . .+ 999×1000 1001! Re-writing sums 1. 100X i=1 tan i 50X i=1 tan i = X 2. NX i=1 (2i 1)5 = N1X i=0 3. ” NX k=1 xk # + ” NX k=0 k xk+1 # = 264 X k= xk 375 + Hint: Write out the sums on the left hand side first, simplify if possible, then write them back into sigma notation. Telescopic sum Calculate the exact value of 137X i=1 1 i 1 i + 1 Hint: Write down the first few terms. Calculate the exact value of 10,000X i=1 1 i(i + 1) Telescopic sum Calculate the exact value of 137X i=1 1 i 1 i + 1 Hint: Write down the first few terms. Calculate the exact value of 10,000X i=1 1 i(i + 1) Warm up: suprema and infima Find the supremum, infimum, maximum, and minimum of the following sets (if they exist): 1. A = [1, 5) 2. B = (1, 6] [ (8, 9) 3. C = {2, 3, 4} 4. D = n1 n : n 2 Z, n > 0 o 5. E = (1)n n : n 2 Z, n > 0 6. F = {2n : n 2 Z} Suprema from a graph Calculate, for the function g on the interval [0.5, 1.5]: 1. supremum 2. infimum 3. maximum 4. minimum Empty set 1. Does ; have an upper bound 2. Does ; have a supremum 3. Does ; have a maximum 4. Is ; bounded above Equivalent definitions of supremum Assume S is an upper bound of the set A. Which of the following is equivalent to “S is the supremum of A” 1. If R is an upper bound of A, then S R . 2. 8R S , R is an upper bound of A. 3. 8R S , R is not an upper bound of A. 4. 8R < S , R is not an upper bound of A. 5. 8R < S , 9x 2 A such that R < x . 6. 8R < S , 9x 2 A such that R x . 7. 8R < S , 9x 2 A such that R < x S . 8. 8R < S , 9x 2 A such that R < x < S . 9. 8" > 0, 9x 2 A such that S ” < x . 10. 8" > 0, 9x 2 A such that S ” < x S . Fix these FALSE statements 1. Let f and g be bounded functions on [a, b]. Then sup of (f + g ) on [a, b] = sup of f on [a, b] + sup of g on [a, b] 2. Let a < b < c . Let f be a bounded function on [a, c ]. Then sup of f on [a, c ] = sup of f on [a, b] + sup of f on [b, c ] 3. Let f be a bounded function on [a, b]. Let c 2 R. Then: sup of (cf ) on [a, b] = c sup of f on [a, b] True or False - Suprema and infima Let A,B ,C R. Assume C A. Which statements are true If possible, fix the false statements 1. IF A is bounded above, THEN C is bounded above. 2. IF C is bounded below, THEN A is bounded below. 3. IF A and C are bounded above, THEN supC supA. 4. IF A and C are bounded below, THEN inf C inf A. 5. IF A and B are bounded, supB supA, and inf A inf B , THEN B A. 6. IF A and B are bounded above, THEN sup(A [ B) = max{supA, supB}. 7. IF A and B are bounded above, THEN sup(A B) = min{supA, supB}. Warm up: partitions Which ones are partitions of [0, 2] 1. [0, 2] 2. {0.5, 1, 1.5} 3. {0, 2} 4. {1, 2} 5. {0, e, 2} 6. {0, 1.5, 1.6, 1.7, 1.8, 1.9, 2} 7. n n + 1 : n 2 N [ {2} Warm up: lower and upper sums Let f (x) = (x 1)2. Consider the partition P = {0, 1, 3} of the interval [0, 3]. Calculate LP(f ) and UP(f ). Lower and upper sums of decreasing functions Let f be a decreasing, bounded function on [a, b]. Let P = {x0, x1, . . . , xN} be a partition of [a, b]. Let xi = xi xi1. What are LP(f ) and UP(f ) Joining partitions Assume LP(f ) = 2, UP(f ) = 6 LQ(f ) = 3, UQ(f ) = 8 1. Is P Q 2. Is Q P 3. What can you say about LP[Q(f ) and UP[Q(f ) The “"–characterization” of integrability True or False Let f be a bounded function on [a, b]. 1. IF “8" > 0, 9 a partition P of [a, b] s.t. UP(f ) LP(f ) < "”, THEN f is integrable on [a, b] 2. IF f is integrable on [a, b] THEN “8" > 0, 9 a partition P of [a, b] s.t. UP(f ) LP(f ) < "”. lower sums upper sums finer partitions finer partitions I ba (f ) I ba (f ) The “"–characterization” of integrability - Part 1 True or False Let f be a bounded function on [a, b]. IF “8" > 0, 9 a partition P of [a, b] s.t. UP(f ) LP(f ) < "”, THEN f is integrable on [a, b] Hints: 1. Recall the definition of “f is integrable on [a, b]”. 2. Let P be a partition. Order the numbers UP(f ), LP(f ), I ba (f ), I b a (f ). (Draw a picture of these numbers in the real line.) The “"–characterization” of integrability - Part 2 True or False Let f be a bounded function on [a, b]. IF f is integrable on [a, b] THEN “8" > 0, 9 a partition P of [a, b] s.t. UP(f ) LP(f ) < "”. Hints: Assume f is integrable on [a, b]. Let I be the integral. Fix " > 0. 1. Recall the definition of “f is integrable on [a, b]”. 2. There exist a partition P1 s.t. UP1(f ) < I + " 2 . Why 3. There exist a partition P2 s.t. LP2(f ) > I ” 2 . Why 4. What can you say about UP1(f ) LP2(f ) 5. Construct a partition P s.t. LP2(f ) LP(f ) UP(f ) UP1(f ). That’s all for today! Problem set 5 is due on Monday July 11 at 11:59 pm. David and Lindsey will substitute me for my oce hours tomorrow. I will be back next week. Good night!