# slower convergent series

###### Theorem.

If

 $\displaystyle a_{1}\!+\!a_{2}\!+\!a_{3}\!+\cdots$ (1)

is a converging series with positive , then one can always form another converging series

 $g_{1}\!+\!g_{2}\!+\!g_{3}\!+\cdots$

such that

 $\displaystyle\lim_{n\to\infty}\frac{g_{n}}{a_{n}}=\infty$ (2)

Proof.  Let $S$ be the sum of (1),  $S_{n}=a_{1}\!+\!a_{2}\!+\cdots+\!a_{n}$  the $n^{\mathrm{th}}$ partial sum of (1) and  $R_{n+1}=S\!-\!S_{n}=a_{n+1}\!+\!a_{n+2}\!+\cdots$  the corresponding remainder term.  Then we have

 $a_{n}=R_{n}\!-\!R_{n+1}=(\sqrt{R_{n}}\!+\!\sqrt{R_{n+1}})(\sqrt{R_{n}}\!-\!% \sqrt{R_{n+1}}).$

We set

 $g_{n}:=\frac{a_{n}}{\sqrt{R_{n}}\!+\!\sqrt{R_{n+1}}}=\sqrt{R_{n}}\!-\!\sqrt{R_% {n+1}}\quad\forall n=1,\,2,\,3,\,\ldots$

Then the series  $g_{1}\!+\!g_{2}\!+\!g_{3}\!+\cdots$  fulfils the requirements in the theorem.  Its $g_{n}$ are positive.  Further, it converges because its $n^{\mathrm{th}}$ partial sum is equal to $\sqrt{R_{1}}\!-\!\sqrt{R_{n+1}}$ which tends to the limit  $\sqrt{R_{1}}=\sqrt{S}$  as  $n\to\infty$  since  $R_{n+1}\to 0$;  this implies also (2).

Title slower convergent series SlowerConvergentSeries 2013-03-22 15:08:24 2013-03-22 15:08:24 pahio (2872) pahio (2872) 11 pahio (2872) Theorem msc 40A05 SlowerDivergentSeries NonExistenceOfUniversalSeriesConvergenceCriterion