Discussion of Theorem $3.25$ of baby Rudin's PMA

Question

Theorem $3.25$ of baby Rudin states as follows.

(a) If $\left|a_n\right|\le c_n$ for $n\ge N_0$, where $N_0$ is some fixed integer, and if $\sum c_n$ converges, then $\sum a_n$ converges.

(b) if $a_n\ge d_n\ge 0$ for $n\ge N_0$, and if $\sum d_n$ diverges, then $\sum a_n$ diverges.

According to part (a), can we say that if the series $\sum a_n$ converges, then we can find a convergent series $\sum c_n$ such that $\left|a_n\right|\le c_n$ for $n\ge N_0$, where $N_0$ is some fixed integer?

The following is Rudin’s Theorem $3.25$. Thanks for your participation.

Answer 1

What you are asking about is the converse to part (a), and the answer is "no." The series $\sum (-1)^n/n$ with $a_n = (-1)^n/n$ converges and satisfies $|a_n|=1/n$. Clearly for any $N_0$ and any series $\sum c_n$ with $c_n\ge |a_n|$ for $n\ge N_0$, the series $\sum c_ n$ diverges.