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

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!

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 $$N_k \le N and note that $$|\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