It's a known thing that the geometric series $ar^{\left(x-1\right)}$ converges, but my question is why? All the proofs I've been able to find online rely on the fact that as $n \rightarrow \infty$, $ar \rightarrow 0$, but wouldn't this be true for the harmonic series $\sum_{n=1}^{\infty}\frac{1}{n}$ which is known to diverge?
2026-03-29 07:29:55.1774769395
Why does the geometric series converge when Harmonic does not?
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