I'm sure this has been asked many times before, sorry if it's a duplicate, but when googling this I mostly found instructions on how to do the integral test.
So I was given an integral $$\int_{1}^{\infty} f(x) dx$$
and asked to find whether it converges/diverges. I figured out that it diverges, and the next task was to find the sum
$$\sum_{n=1}^{\infty} f(n) $$ where $f(x)$ was the same expression in both tasks. So I concluded that since the integral diverges, the sum also diverges by the integral test. But assume I was given the sum of the series first, and let's say I were to use a limit-comparison test to figure out that the series diverges. If the next task was to calculate the integral, would it still hold to conclude that since the sum of the series $a_n$ diverges, then so does the integral, or does the implication not go both ways? I hope my question was clear, thanks in advance.
edit: Original $f(x)$ was $arctan(\frac{1}{x})$ and $f(n) = arctan(\frac{1}{n})$
Answer
If you let $f(x) = \sin(\pi x)$ you have a divergent integral but a convergent sum.
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