The limit of a divergent series

Question

Let $\{a_n\}_{n \in \mathbb{N}} \subset \mathbb{R_+}$ be a real positive sequence such that $$ \sum_{n=1}^\infty a_n =\infty $$ a I would like to konw if is it true that:

$$ \lim_{k \to \infty} \sum_{n=k+1}^\infty a_n =0 $$

Thanks.

Answer 1

No. Take $a_n=n$. Then

$$\sum_{n=1}^\infty a_n=\infty \qquad\text{and}\qquad \sum_{n=k+1}^\infty a_n=\infty$$

for all $k\in\Bbb N$. Actually this is the rule and no exception. For general $a_n$ we have

$$\sum_{n=k+1}^\infty a_n=\sum_{n=1}^\infty a_n-\sum_{n=1}^k a_n=\infty-\sum_{n=1}^k a_n = \infty$$

because the latter sum is a finite number.

Answer 2

No. Take $a_n=1$ for each natural $n$. Indeed, you can even take $a_n=\frac1n$, and this exemple shows that your statement is false even if we assume that $\lim_{n\to\infty}a_n=0$.

Answer 3

No. If $$ \sum_{n=1}^\infty a_n =\infty $$ then $$ \sum_{n=k+1}^\infty a_n =\infty $$ for all $k$. Thus, $$ \lim_{k \to \infty}\sum_{n=k+1}^\infty a_n =0 $$ is false.