If each term $a_k \geq 0$, then the series $\sum_{k=i}^{\infty}a_k$ is summable if and only if the sequence of a partial sums is bounded.
Proof: $\rightarrow$ Let $S_n$ be a partial sums
$$S_n = \sum_{k=i}^{\infty}a_k$$. such series is is of the form of a monotone sequence since $a_k \geq 0$ thus $S_n$ is bounded since every monotone sequence converges. and we kn0w that converges implies bounded.
I do not know for sure if this is reasonable. and also i do not know how to go to the other direction any idea that would be appreciated. thanks
Best Answer
Assume $S_n < M, \forall n \geq 1$, then $S_n$ is an increasing sequence and is bounded above by $M$, hence converges to $L$. Thus $L = \displaystyle \lim_{n\to \infty} S_n = \displaystyle \sum_{i=1}^{\infty} a_i$. Thus the series is summable. Conversely, if the series is summable, and call the sum is $L$, then $L \geq S_n, \forall n \geq 1$, thus $\{S_n\}$ is bounded above by $L$.