Let \(f\) be a positive, continuous, decreasing function such that \(a_n=f(n)\). If \(\displaystyle \sum_{n=1}^\infty a_n\) converges to \(k\), which of the following must be true?

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The integral test states that for a positive, continuous, and decreasing function,

$$ \sum_{n=1}^\infty a_n \hskip{2em} \text{ and } \hskip{2em}\int\limits_1^\infty f(x)   dx $$

either both converge or diverge.

The test does not tell us the value the integral or series converges to.