I know that in order to apply the integral test for convergence or divergence a function $f(x)$ must be positive, continuous, and decreasing. However, I was wondering if $$ \int_{1}^{\infty}f(x)\, dx\, =\, \sum_{n=1}^{\infty}f(x) $$
because finding the integral of the function would essentially be finding the sum of the function in series itself?
Thank you!
No, they are not equal. Consider the function $f(x)=\dfrac{1}{x^2}$. We have $$\int_1^\infty\frac{1}{x^2}\, dx=1$$ but $$\sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}$$