For \(x\geq 1\), the continuous function \(g\) is decreasing and positive. A portion of the graph of \(g\) is shown above. For \(n\geq1\), the \(n\)th term of the series \(\displaystyle \sum_{n=1}^\infty a_n\) is defined by \(a_n=g(n)\). If \(\displaystyle \int\limits_1^\infty g(x)   dx\) converges to 8, which of the following could be true?

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By the integral test, both the improper integral and series converges.

The value of the series is:

$$ \sum_{n=1}^\infty a_n = g(1) + g(2) +g(3)+...+g(n) $$

This is equivalent to a left riemann sum. The area under the curve is given as 8, so the sum of the series must be greater than 8. Only the third choice satisfies this.