Decreasing sequence term sign for zero convergent series Given a convergent series 
$$\lim_{n \to \infty} \sum_{k=1}^n a_k = 0 $$ for an decreasing sequence $\{a_n\}$. Can we make any deductions on the sign of sequence terms $a_n$? 

If $(a_n)_{n \in \mathbb{Z}^+}$ is decreasing and $\sum_{k=1}^\infty a_k = 0$, then $a_n = 0$ $\forall n$.
To see this, we first show that $a_n \geq 0$. Suppose not, so $a_1 = q < 0$. Then since the sequence is decreasing, $a_n \leq q$ $\forall n$. Then:
$$
\sum_{k=1}^\infty a_k \leq \sum_{k=1}^\infty q = -\infty
$$
Which is not possible. 
We then observe that $a_1 = 0$ necessarily. This is because if $a_1 > 0$, then since $a_n \geq 0$ $\forall n$, we have that:
$$
\sum_{k=1}^\infty a_k \geq a_1 > 0
$$
Which is, again, a contradiction.
This means that we have $0 = a_1 \geq a_2 \geq a_3 \geq \cdots \geq 0$. The only possibility is that $a_n = 0$ $\forall n$.
