In my Calculus text book there is a theorem named The Integral Test that states:
If the function $f$ is continous, positive and nondecreasing on the interval $x\in[a, \infty)$, then
$$\sum^{\infty}_{x=1} f(x) \text{ converges} \Leftrightarrow \int^{\infty}_{a} f(x) \text{ converges}$$
Based on my home made proof, the more general result
$$\sum^{b}_{x=1} f(x) \text{ converges} \Leftrightarrow \int^{b}_{a} f(x) \text{ converges}$$
should also hold.
Is my result true or is indeed The Integral Test limited to series and improper integrals?
Better write $\sum^{\infty}_{k=1} f(k)$ instead of $\sum^{\infty}_{x=1} f(x)$.
If $b \in \mathbb N$, then $\sum^{b}_{k=1} f(k)$ is a finite sum ! No convergence considerations are needed !
What do you mean by "$\int^{b}_{a} f(x) dx \text{ converges}$" ?