TITLE: Infinite sum conundrum: $S_1 = 1 + 2 + \ldots$, $S_2 = \frac{1}{2} + 1 + 2 + \ldots$, $S_1-S_2$ finite or infinite?
QUESTION [5 upvotes]: $$S_1 = 1 + 2 + 3 + \ldots$$
$$S_2 = \frac{1}{2} + 1 + 2 + 3 + \ldots$$
Now case 1:
$$\left.\Delta S = S_1 - S_2 = \frac{1}{2}\,\right\}\text{finite value}$$
Case 2:
$$\Delta S = S_1 - S_2 = \frac{1}{2} + \underbrace{(1 + 1 + 1 + \ldots)}_{\infty\text{ value!}}$$
In one case, I see a finite  difference whereas in the other, an infinite. So, which one is right and why?

REPLY [7 votes]: If $\lim_{n \to \infty}u_n$ and $\lim_{n \to \infty}v_n$ exist and are finite then

$$
\lim_{n \to \infty}(v_n-u_n)=\lim_{n \to \infty}v_n-\lim_{n \to \infty}u_n. \tag1
$$ 

You seem to have applied $(1)$ to
$$
u_n=1+2+\cdots+n, \quad v_n=\frac12+1+2+\cdots+n
$$ which is not justified since
$$
\lim_{n \to \infty}u_n=\infty,\qquad \lim_{n \to \infty}v_n=\infty.
$$