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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 $a_n$?

Thoth
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1 Answers1

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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 $a_n = 0$ $\forall n$.

Clement Yung
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  • (+1) You saved me some time. But I suspect that the OP will change the question. I hope that I am wrong. – José Carlos Santos Feb 11 '20 at 08:46
  • Generally, I know that if $$\lim_{n \to \infty} \sum_{k=1}^n a_k = 0 $$ then $$\lim_{n \to \infty}a_n = 0.$$ So if the sequence is decreasing then all sequence terms must equal to zero? [Sorry for my naive comment but I am not familiar with sequences and series] – Thoth Feb 11 '20 at 09:04
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    You don't need $\sum_{k=1}^\infty a_k$ to be zero. As long as the sum converges, you can conclude that $a_n \to 0$ (See here). In this case, if you restrict further that 1) the sum to be zero and 2) $(a_n)$ is decreasing, then all terms will be zero. – Clement Yung Feb 11 '20 at 09:08
  • Thanks again things are more clear! Please let me ask something last hoping it is not out of the scope. Alternatively, If the sequence $a_n$ is increasing we also conclude to $a_n=0~\forall n$? – Thoth Feb 11 '20 at 09:44
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    Yes we can. We can just reverse all the inequalities in my solution. – Clement Yung Feb 11 '20 at 10:03