quiz (120) - Quiz 8 Section 4 9:05-9:55 recitation Consider...

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Quiz 8 — March 16, 2011 – Section 4 – 9:05-9:55 recitation. . Consider the series n =2 2 n 2 - 1 . (a) Let M 2 be some fxed integer. Find a closed ±ormula ±or the partial sum s M = M n =2 2 n 2 - 1 . (Comment. It is possible to use the technique o± partial ±ractions to express this series as a telescoping series.) (b) What is the sum o± the series? Answer. Write 2 n 2 - 1 = A n - 1 + B n +1 . Multiply by n 2 - 1 to see that 2 = A ( n + 1) + B ( n - 1) . Plug in n = 1 to see that 1 = A . Plug in n = - 1 to see that B = - 1. This works becasue 1 n - 1 + - 1 n + 1 = n + 1 - ( n - 1) n 2 - 1 = 2 n 2 - 1 . (a) We see that s M = M s n =2 2 n 2 - 1 = M s n =2 b 1 n - 1 - 1 n + 1 B = ± 1 2 - 1 - 1 2+1 ² + ± 1 3 - 1 - 1 3+1 ² + ± 1 4 - 1 - 1 4+1
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