sequence determining convergence of series

Theorem.  Let $a_1+a_2+\mathrm{\dots }$ be any series of real $a_n$.  If the positive numbers $r_1,r_2,\mathrm{\dots }$  are such that

$\underset{n\to \mathrm{\infty }}{lim}{\displaystyle \frac{a_n}{r_n}}=L\ne \mathrm{\hspace{0.17em}0},$(1)

then the series converges simultaneously with the series $r_1+r_2+\mathrm{\dots }$

Proof.  In the case that the limit (1) is positive, the supposition implies that there is an integer $n_0$ such that

(2)

Therefore

and since the series ${\sum }_{n=1}^{\mathrm{\infty }}0.5L{r}_n$ and ${\sum }_{n=1}^{\mathrm{\infty }}1.5L{r}_n$ converge simultaneously with the series $r_1+r_2+\mathrm{\dots }$, the comparison test guarantees that the same concerns the given series $a_1+a_2+\mathrm{\dots }$

The case where (1) is negative, whence we have

$$\underset{n\to \mathrm{\infty }}{lim}\frac{-a_n}{r_n}=-L>0,$$

may be handled as above.

Note.  For the case  $L=0$, see the limit comparison test.