Question about convergence and divergence of positive series,which means there exists no slowest convergent series.

Question

• If positive series $\sum_{n=1}^{\infty} a_n$ is divergent,try to prove exist divergence positive series $\sum_{n=1}^{\infty} b_n$,which satisfies
  $\lim _{n \rightarrow \infty} \frac{b_n}{a_n}=0$
• If positive series $\sum_{n=1}^{\infty} a_n$ is convergent,try to prove exist convergence positive series $\sum_{n=1}^{\infty} b_n$,which satisfies $\lim _{n \rightarrow \infty} \frac{a_n}{b_n}=0$
• Also I wonder,about positive series $\sum_{n=1}^{\infty} a_n$,consider its sum sequence $S_n=\sum_{k=1}^{n} a_k$.
  If $\sum_{n=1}^{\infty} a_n$ is divergent,how about the $\sum_{n=1}^{\infty} S_n$
  It's also divergent?Or not？
  If $\sum_{n=1}^{\infty} a_n$ is convergent,how about the $\sum_{n=1}^{\infty} S_n$
  It's also convergent?Or not？

Answer 1

Let $S_n=\sum_{i=1}^na_i$ and suppose $a_n>0$, we have the following results.

Question 1:

• If $a_n\!\not\to\!0$ there exist $\varepsilon>0$ and an increasing sequence $(n_1,n_2,...)$ such that $a_{n_i}\geq\varepsilon$ for all $i$. $$\text{Let $b_n=\varepsilon/i$ if $n\in(n_i)$ and $b_n=\min(a_n^2,1/n^2)$ otherwise,}$$ then $\sum_nb_n\geq\varepsilon\sum_n1/n\to\infty$ and $b_n/a_n\leq1/n\to0$.

• If $a_n\to0$, let $b_n=\ln(S_{n+1}/S_n)$, then $\sum_{i=1}^nb_i=\ln(S_{n+1}/S_1)\to\infty$ and $$\frac{b_n}{a_n}=\frac1{a_n}\ln\left(\frac{S_n+a_{n+1}}{S_n}\right)=\frac{a_{n+1}}{a_n}\ln\left(1+\frac{a_{n+1}}{S_n}\right)^{\frac1{a_{n+1}}}.$$ As $n\to\infty$, we have $a_{n+1}/S_n\to0$, so by the limit definition of $e$ it follows $(1+a_{n+1}/S_n)^{1/a_{n+1}}\to e^0=1$, thus $b_n/a_n\to0$ since $a_{n+1}/a_n\to1$.

Question 2:

• Suppose $S_n\to S$ then let $b_n=\sqrt{S-S_n}-\sqrt{S-S_{n+1}}$, it follows $\sum_nb_n\to\sqrt{S-S_1}$ and $$\frac{a_n}{b_n}=\frac{a_n}{\sqrt{S-S_n}-\sqrt{S-S_{n+1}}}=\frac{a_n}{a_{n+1}}\big(\sqrt{S-S_n}+\sqrt{S-S_{n+1}}\big)\to0.$$

Question 3:

• $a_n>0\ \Rightarrow\ S_n\!\not\to0\ \Rightarrow\ \sum_nS_n$ diverges.