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So I have read that series converge when a limit exists, ant it does diverge if there is no limit or it goes to infinity.
So then I saw this example, where I have series:
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And then 1) When a > 1 they say it converges, well which I understand why, the limit then is 0.
And then I see 2) When 0 < a <= 1, and they say this diverges?? Why does this diverge if the limit is 1? The limit exists and it's not infinity, then why does it diverge? I am getting confused.

Norbiuxx
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    You are conflating $\lim_{n \to \infty} \frac{1}{n^\alpha}$ and $\sum_{n=1}^\infty \frac{1}{n^\alpha}$. The former limit being zero does not imply convergence of the latter series. – angryavian May 25 '21 at 19:26
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    The series $\sum_{n=1}^\infty \frac1{n^\alpha}$ converges for values of $\alpha>1$ and it diverges for values of $\alpha\le 1$. The general term of the series $\frac1{n^\alpha}$ decreases to $0$ for $\alpha>0$, is equal to $1$ for $\alpha=0$, and approaches $\infty$ for $\alpha<0$. – Mark Viola May 25 '21 at 19:27
  • (1) "A series is convergent if the sequence of its partial sums tends to a limit." (2) You are only looking at the last term from which we can only deduce divergence if it's not equal to zero. (3) In order to prove convergence apply the integral test. – vitamin d May 25 '21 at 19:27
  • Of particular note, is the harmonic series, $\sum_{n=1}^{\infty} (1/n)$ which is known to be divergent. In fact, this Wikipedia article discusses that series. – user2661923 May 25 '21 at 19:46

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