sum of the inverses

In real arithmetic, the sum of the inverses of all positive integers goes to infinity. In other words, if we take enough terms in the sum the sum exceeds any fixed value.
However, when we compute the sum in Java's floating point arithmetic the largest value we get is 15.403683, regardless of how many terms we add. Explain the possible reasons for this anomaly.

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...sum that, in Java's floating point system, sum+1/n=sum because of loss of ...