**Lemma**: Given a [[Real sequences|sequence]] $(a_{n})$. If $a_{n}\geq 0$ then $\sum_{n=1}^{\infty} a_{n}$ [[Convergent Real Series|converges]] $\iff$ it's partial sums are [[Bounded Sequence|bounded]] above. **Proof**: [[Monotone Bounded Real Sequence is Convergent|Monotone convergence theorem]]. **Examples** 1. [[Harmonic Numbers]] is a series with positive terms that is not bounded above so it does not converge. 2. [[Basel Problem]]. ### References - [[Integral test for convergence of series of non-negative decreasing function]]. - [[Every rearrangement of a series with positive terms has the same limit]].