Suppose $f(x)$ is a positive, continuous, and decreasing function such that $\int_{100}^\infty f(x) dx $ is finite. Which of the following statements about $\sum_{n=0}^\infty f(n)$ is true?
$(a)$ It must converge.
$(b)$ It must diverge.
$(c)$ It might either converge or diverge; we need more information to know for sure.
I think the series converges because a series is just a constant plus the other one. Therefore, if one converges, so does the other. (So the answer above would be $(a)$. Any guidance would be appreciated.
The integral $\int_0^{\infty} f(x) dx = \int_0^{100} f(x)dx + \int_{100}^{\infty} f(x)dx$ is convergent because the two terms are finite: the first one is finite because the function is continuous so by the extreme value theorem it attains a maximum and a minimum on $[0,100]$. So by integral test for convergence also the series $\sum_{n=0}^\infty f(n)$ is convergent.