slower convergent series

Theorem.

If

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

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

$$g_1+g_2+g_3+\mathrm{\cdots }$$

such that

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

Proof.  Let $S$ be the sum of (1),  $S_n=a_1+a_2+\mathrm{\cdots }+a_n$  the $n^{\mathrm{th}}$ partial sum of (1) and  $R_{n+1}=S-S_n=a_{n+1}+a_{n+2}+\mathrm{\cdots }$  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

$$g_n:=\frac{a_n}{\sqrt{R_n}+\sqrt{R_{n+1}}}=\sqrt{R_n}-\sqrt{R_{n+1}}\mathit{\hspace{1em}}\forall n=1,\mathrm{\hspace{0.17em}2},\mathrm{\hspace{0.17em}3},\mathrm{\dots }$$

Then the series  $g_1+g_2+g_3+\mathrm{\cdots }$  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  $n\to \mathrm{\infty }$  since  $R_{n+1}\to 0$;  this implies also (2).