Prove that if n > 1, the nth partial sum of the harmonic series is not an integer. Hint: Let 2 k be the largest power of 2 that is less than or equal to n and let M be the product of all odd integers that are less than or equal to n . Suppose that s n = m , an integer. Then M 2 k s n = M 2 k m . The right side of this equation is even. Prove that the left side is odd by showing that each of its terms is an even integer, except for the last one.
Prove that if n > 1, the nth partial sum of the harmonic series is not an integer. Hint: Let 2 k be the largest power of 2 that is less than or equal to n and let M be the product of all odd integers that are less than or equal to n . Suppose that s n = m , an integer. Then M 2 k s n = M 2 k m . The right side of this equation is even. Prove that the left side is odd by showing that each of its terms is an even integer, except for the last one.
Solution Summary: The author proves that the nth partial sum of the harmonic series is not an integer.
Prove that if n > 1, the nth partial sum of the harmonic series is not an integer. Hint: Let 2k be the largest power of 2 that is less than or equal to n and let M be the product of all odd integers that are less than or equal to n. Suppose that sn = m, an integer. Then M2ksn = M2km. The right side of this equation is even. Prove that the left side is odd by showing that each of its terms is an even integer, except for the last one.
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