$\sum_{n=0}^{+\infty}a_{n}$ converges $\Leftrightarrow $ $\forall k \geqslant 0$, $a_{k}>\sum_{n=k+1}^{+\infty}a_{n}$

Question

Is it true that

$\sum_{n=0}^{+\infty}a_{n}$ converges $\Leftrightarrow $ $\forall k \geqslant 0$, $a_{k}>\sum_{n=k+1}^{+\infty}a_{n}$

with $a_{n}\geqslant 0$ ?

Answer 1

No, it is not true. Suppose that$$a_n=\begin{cases}1&\text{ if }n=2\\0&\text{ otherwise.}\end{cases}$$Then $\sum_{n=0}^\infty a_n$ converges, but $0=a_1<\sum_{n=2}^\infty a_n=1$.

Answer 2

No. $a_n = \frac{1}{n^2}$ fails since $a_n < a_{n+1}+a_{n+2}$ for $n \ge 4$.

Answer 3

If $a_{k}>\sum_{n=k+1}^{+\infty}a_{n} $ for any $k$ then the sum converges since $\sum_{n=1}^{k}a_{n} $ is bounded.