Problem:
Let A be the set of positive integers that have no prime factors other than 2,3, or 5. The infinite sum
11​+21​+31​+41​+51​+61​+81​+91​+101​+121​+151​+161​+181​+201​+⋯
of the reciprocals of all the elements of A can be expressed as nm​, where m and n are relatively prime positive integers. What is m+n?
Answer Choices:
A. 16
B. 17
C. 19
D. 23
E. 36
Solution:
Elements of set A are of the form 2iâ‹…3jâ‹…5k for nonnegative integers i,j, and k. Note that the product
(1+21​+221​+⋯)(1+31​+321​+⋯)(1+51​+521​+⋯)
will produce the desired sum. By the formula for infinite geometric series, this product evaluates to
1−21​1​⋅1−31​1​⋅1−51​1​=2⋅23​⋅45​=415​
The requested sum is 15+4=19​.
The problems on this page are the property of the MAA's American Mathematics Competitions