sequence determining convergence of series

Theorem.  Let $a_{1}\!+\!a_{2}\!+\ldots$ be any series of real $a_{n}$.  If the positive numbers $r_{1},\,r_{2},\,\ldots$  are such that

 $\displaystyle\lim_{n\to\infty}\frac{a_{n}}{r_{n}}\;=\;L\;\neq\,0,$ (1)

then the series converges simultaneously with the series $r_{1}\!+\!r_{2}\!+\ldots$

Proof.  In the case that the limit (1) is positive, the supposition implies that there is an integer $n_{0}$ such that

 $\displaystyle 0.5L\;<\;\frac{a_{n}}{r_{n}}\;<\;1.5L\quad\textrm{for }n\geqq n_% {0}.$ (2)

Therefore

 $0\;<\;0.5Lr_{n}\;<\;a_{n}\;<\;1.5Lr_{n}\quad\textrm{for all }n\geqq n_{0},$

and since the series $\sum_{n=1}^{\infty}0.5Lr_{n}$ and $\sum_{n=1}^{\infty}1.5Lr_{n}$ converge simultaneously with the series $r_{1}\!+\!r_{2}\!+\ldots$, the comparison test guarantees that the same concerns the given series $a_{1}\!+\!a_{2}\!+\ldots$

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

 $\lim_{n\to\infty}\frac{-a_{n}}{r_{n}}\;=\;-L>0,$

may be handled as above.

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

Title sequence determining convergence of series SequenceDeterminingConvergenceOfSeries 2013-03-22 19:06:54 2013-03-22 19:06:54 pahio (2872) pahio (2872) 6 pahio (2872) Definition msc 40A05 LimitComparisonTest