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Problem

Show that $1 +\frac{1}{2} +\frac{1}{3}+ \cdots+ \frac{1}{n}$ can be made larger than any real number by choosing an appropriate $n$. Then show that when all the terms with denominators that are not prime are removed, the series diverges. In other words the sum of the reciprocals of the prime numbers diverges. On the other hand, show when all the terms that contain the digit nine are deleted, the series converges. In fact, the sum is less than 90.

Details
Contributer: Tom R
Authors
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References
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Problem Sets This Problem Belongs to:
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VARIABLES
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DEFINITIONS
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