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Unformatted text preview: x 6 = 1, then x + 2 x 2 + 3 x 3 + ··· + nx n = x-( n + 1) x n +1 + nx n +2 (1-x ) 2 for every positive integer n . Question 4. Let H n represent the partial sum of the Harmonic series: H n = 1 + 1 2 + 1 3 + 1 4 + ··· + 1 n . Show that H 2 k ≤ 1 + k for all k ≥ 0. Question 5. In the mound-splitting game, we start oﬀ with a single mound of pebbles. In each move, we pick a mound, split it into two smaller mounds of arbitrary size (say, k and m pebbles), multiply the number of pebbles in the two mounds and write down the result (that is, k × m ). We continue playing until every mound has only one pebble (for which we write down the number 0). At the end, we add up all the numbers written down after the splits. Prove that if we start with n pebbles, then the ﬁnal sum is n ( n-1) / 2 irrespective of how we split the mounds, or in which order we split them . 1...
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This note was uploaded on 09/28/2011 for the course ECE 103 taught by Professor Nayak during the Spring '11 term at Waterloo.
- Spring '11