Proximity to perfect squares

Calculus Level 1

A function \(f(n)\) is defined over positive integers as follows:

\[f(n) = \begin{cases} 0 & \text{if }n\text{ is a perfect square}; \\ 1 & \text{if }n\text{ is closer to the perfect square before it than to the one after it}; \\ -1 & \text{otherwise}. \end{cases}\]

For example, \(f(1) = 0\) because 1 is a perfect square; \(f(2) = 1\) because 2 is closer to 1 than it is to 4; \(f(7) = -1\) because 7 is closer to 9 than it is to 4.

What can be said about the series \(\displaystyle \sum_{n=1}^{\infty} \frac{f(n)}{n}?\)

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