I will write down a brief "lemma" so to say which I've learned from Piskunov's Calculus. Which generalizes this:
Let $y=f(x)$ be a function. Let $a_n = f(n)$ and $s_n = \displaystyle \sum_{k=1}^n a_n$. Then
$\int_1^{n+1} f(x) dx < s_{n+1} < a_1+\int_1^{n+1}f(x)dx$
Then $s_n$ converges only if the integral converges, and
$\int_1^{\infty} f(x) dx < s < a_1+\int_1^{\infty}f(x)dx$
In your case you have $a_1 = 1$, thus you have $1 < s < 2$, but the improper integral is smaller than the sum, as you can see from the lemma.
This can be derived by simply inspectioning the partial sums as rectangles of height $f(k)$ and width $1$, and comparing the upper and lower sums and the actual integral to the value of the series. Some graphing is all it takes.