To decompose a conditionally convergent series into a partial bounded series and another decreasing series

Question

In the first-year mathematics analysis course, the instructor assigned a problem on the convergence of series. We are given that a series $\sum_{n=1}^\infty A_n$ converges absolutely if $\sum_{n=1}^\infty |A_n|<+\infty$. Alternatively, it converges conditionally if its partial sums converge, while $\sum_{n=1}^\infty |A_n|=+\infty$. The problem is stated as follows:

Suppose $\sum_{n=1}^\infty A_n$ converges. The task is to prove the existence of sequences ${a_n}$ and ${b_n}$ such that $A_n=a_nb_n$ for $n\geq 1$. At the same time, the partial sum of $\sum_{n=1}^\infty a_n$ must be bounded, while ${b_n}$ is monotonically decreasing and tends to $0$.

I have successfully solved the case when $\sum_{n=1}^\infty A_n$ converges absolutely. In this scenario, we define $R_n=\sum_{k\geq n}|A_k|$ and we construct $$ b_n:=\sqrt{R_{n+1}}+\sqrt{R_n},\quad a_n:=\frac{A_n}{b_n}. $$ These sequences satisfy the given requirements. However, I am facing difficulty in solving the case when $\sum_{n=1}^\infty A_n$ converges conditionally. I am hopeful that someone can provide assistance. Thank you very much!

Answer 1

The idea here is that $\sum_{N \le n \le M}A_n \to 0, M,N \to \infty$ so fix a strictly increasing sequence $N_k, k \ge 1$ st $$|\sum_{N \le n \le M}A_n| \le \frac{1}{k^2}, M,N \ge N_k$$ where of course you can use other absolutely convergent series instead of $\sum 1/k^2$ with appropriate choices as below.

Then for $N_k \le n < N_{k+1}$ pick $a_n=A_n \sqrt k, b_n =1/\sqrt k$ (while if you want $b_n$ strictly decreasing you can wiggle it with a very small strictly decreasing sequence depending on $\max_{n < N_{k+1}}|A_n|$)

Then for any $N \ge N_1$ pick $k$ st $N_k \le N <N_{k+1}$ and note that $$|\sum_{n \le N}a_n| \le |\sum_{1 \le n < N_1}a_n|+|\sum_{N_1 \le n < N_2}a_n|+...|\sum_{N_k \le n \le N}a_n|$$

But now $|\sum_{n \le N_1}a_n|=A$ and each $|\sum_{N_r \le n < N_{r+1}}a_n|=\sqrt r |\sum_{N_r \le n < N_{r+1}}A_n| \le \frac{1}{r^{3/2}}$ by our choices.

Hence the partial sums of $a_n$ are bounded by $A+\sum_{r \ge 1}\frac{1}{r^{3/2}}$ for $N \ge N_1$ and of course the first few up to $N_1$ are bounded by some other constant $A_1$ hence they are all bounded uniformly and we are done