# Is the integral test necessary and sufficient for convergence?

So, I was reading about the integral test for convergence of a series from wikipedia, which says that the series converges if the integral of the monotonically decreasing function converges, but to my intuition

$$\int_{1}^{\infty}f(x)\,\mathrm{d}x\leq\sum_{k=1}^{+\infty}f(k)$$

Simply because the function is monotonically decreasing and positive. In that case I might have integrals converging and the series diverging, all I would be able to say is that if the integral diverges, the series definitely diverges. I mean, it looks like a test of divergence more than a test of convergence. Am I going wrong somewhere? Any help is sincerely appreciated since I am not one with a formal mathematical background.

• No capital letters, please. – Saad Jun 9 '18 at 9:30
• @AlexFrancisco Ok, no problems....someone has already edited it – ubuntu_noob Jun 9 '18 at 9:32

Note that for any $N$, we have
$$\sum_{n=2}^{N} f(n) \leq \int_1^N f(x) \; dx\leq \sum_{n=1}^{N} f(n).$$
Note that since $f$ is decreasing $$\int_{1}^{\infty}f(x)dx \ge \int_{1}^{\infty}f(\lceil x \rceil)dx = \sum_{n=2}^{\infty}f(n).$$