slower divergent series

Theorem.

If

$a_1+a_2+a_3+\mathrm{\cdots }$ (1)

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

$$s_1+s_2+s_3+\mathrm{\cdots }$$

with positive such that

$\underset{n\to \mathrm{\infty }}{lim}{\displaystyle \frac{s_n}{a_n}}=0.$ (2)

Proof.   Let  $S_n=a_1+a_2+\mathrm{\cdots }+a_n$  be the $n^{\mathrm{th}}$ partial sum of (1).  Then we have

$$a_n=S_n-S_{n-1}=(\sqrt{S_n}+\sqrt{S_{n-1}})(\sqrt{S_n}-\sqrt{S_{n-1}}).$$

We set  $s_1:=\sqrt{S_1}$  and

$$s_n:=\frac{a_n}{\sqrt{S_n}+\sqrt{S_{n-1}}}=\sqrt{S_n}-\sqrt{S_{n-1}}$$

for  $n=2,\mathrm{\hspace{0.17em}3},\mathrm{\hspace{0.17em}4},\mathrm{\dots }$  Then the of the series

$$\sum _{n=1}^{\mathrm{\infty }}{s}_n=\sqrt{S_1}+\sum _{n=1}^{\mathrm{\infty }}(\sqrt{S_{n+1}}-\sqrt{S_n})$$

apparently are positive.  This series is however divergent, because the sum of its $n$ first is equal to $\sqrt{S_n}$ which grows without bound along with $n$ since (1) diverges.  For this reason we also get the result (2).

Remark.  Niels Henrik Abel has presented a simpler example on such series $s_1+s_2+s_3+\mathrm{\cdots }$:

$$1+\frac{a_2}{a_1+a_2}+\frac{a_3}{a_1+a_2+a_3}+\frac{a_4}{a_1+a_2+a_3+a_4}+\mathrm{\cdots }$$

Title	slower divergent series
Canonical name	SlowerDivergentSeries
Date of creation	2013-03-22 15:08:27
Last modified on	2013-03-22 15:08:27
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	13
Author	pahio (2872)
Entry type	Theorem
Classification	msc 40A05
Related topic	SlowerConvergentSeries
Related topic	NonExistenceOfUniversalSeriesConvergenceCriterion