Necessary condition of changing signs of a divergent series $\sum_{n=1}^{\infty}p_{n}$ to make it convergent,$p_{n}$ decreases and tends to $0$.

Let $S_n = \epsilon_1 + \ldots + \epsilon_n$, and apply summation by parts to get$$\sum_{j=1}^n \epsilon_jp_j = S_n p_n + \sum_{j=1}^{n-1} S_j(p_j - p_{j+1}).$$

If $\liminf S_n/n = \beta > 0$, then there exists $N$ such that $S_n/n > \beta/2$ for all $n > N$. Using the hypothesis that $p_n$ decreases and is positive since it converges to $0$ we have that $p_j - p_{j+1} > 0$ for all $j$ and

$$\begin{align}\sum_{j=1}^n \epsilon_jp_j &> \frac{\beta}{2}np_n + \sum_{j=1}^{N} S_j(p_j - p_{j+1}) + \frac{\beta}{2} \sum_{j=N+1}^{n-1}j(p_j - p_{j+1}) \\ &= \frac{\beta}{2}np_n + \sum_{j=1}^{N} S_j(p_j - p_{j+1}) + \frac{\beta}{2} \sum_{j=N+1}^{n-1}(jp_j - (j+1)p_{j+1}) + \frac{\beta}{2}\sum_{j=N+1}^{n-1}p_{j+1} \end{align}$$

The second sum on the RHS is telescoping and, thus,

$$\tag{*}\begin{align}\sum_{j=1}^n \epsilon_jp_j &> \sum_{j=1}^{N} S_j(p_j - p_{j+1}) + \frac{\beta}{2}(N+1)p_{N+1} + \frac{\beta}{2}\sum_{j=N+1}^{n-1}p_{j+1}\end{align}$$

With $N$ fixed, the first two terms on the RHS of (*) remain constant, but the last sum tends to $+\infty$ as $n \to \infty$ since $\sum p_n$ diverges. This contradicts the convergence of $\sum \epsilon_n p_n$.

Thus, $\liminf (\epsilon_1 + \ldots + \epsilon_n)/n \leqslant 0$.

By a similar argument we can show that $\limsup (\epsilon_1 + \ldots + \epsilon_n)/n \geqslant 0$.