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I am interested on the behavior of the series $$\lim_{n \to \infty} \sum_{k=1}^n a_k < 0 $$ with $\lim_{n \to \infty} a_n = a^* $. Can we say that this series converges? Also, is it possible to define any condition on $a_k$ (e.g., decreasing sequence) in order to conclude that $a_k <0$? Sorry for my naive question I am not experienced with series.

Thoth
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  • The question is clear but I’m afraid that it will be deemed naive and be voted to close... – Szeto Feb 11 '20 at 15:49
  • If the limit exists, the series is said to verge, whatever the sign of the limit. As to the sign of its terms, I know of no obvious answer. – Bernard Feb 11 '20 at 15:51
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    If the limit $\lim_{n\rightarrow\infty}\sum_{k=1}^{n}a_k$ exists ( positive or negative) then it is easy to see that $\lim_{k\rightarrow\infty}a_k=0$. – Peter Melech Feb 11 '20 at 16:00

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When you write $\lim_{n \to \infty} \sum_{k=1}^n a_k < 0$ (or, as it is usually written, $\sum_{k=1}^\infty a_k < 0$, you're assuming that the limit exists. In other words, the sum indeed converges. One can show that if the sum converges, then $\lim_{n \to \infty} a_n = 0$ necessarily.

However, the converse is not true, i.e. to $\lim_{n \to \infty} a_n = 0$, then $\sum_{k=1}^\infty a_k$ may or may not converge. The most classic and famous example is the harmonic series, where $a_n = \frac{1}{n}$. Clearly $a_n = \frac{1}{n} \to 0$, but $\sum_{k=1}^\infty \frac{1}{n}$ diverges to $+\infty$.

Clement Yung
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  • Thanks for you interest. Can we make any deduction on the sign of $a_k$ if we know that $a_k$ is an increasing or decreasing sequence? In other words, can we know if $a_k$ approaches zeros from the negative or positive? – Thoth Feb 11 '20 at 16:14
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    If $a_k$ is increasing or decreasing (not necessarily strict), then we must have $a_k \leq 0$ and $a_k \geq 0$ necessarily. As a result, if $a_k$ is increasing then $a_k \to 0$ from the negative sign (and the opposite if $a_k$ is decreasing). – Clement Yung Feb 11 '20 at 16:17