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Why the solution says $\sum_{n=1}^\infty \dfrac{1}{n^{1.5}}$ converges? Does every series $\sum_{n=1}^\infty \dfrac{1}{n^{x}}$ converges to 0 except $1/n$ (harmonic Series)?

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I found that after verifying a series with series convergence test, especially for comparison test and limit comparison test, I do not have a clear mind on verifying the series that I comparing is converge or diverge. Do I need to use the partial sum to test the convergence of series that I comparing every time?

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    $\begingroup$ For $\sum_1^\infty \frac{1}{n^{1.5}}$, we can show convergence using the Integral Test (or Cauchy Condensation). After we have shown (same way) that $\sum \frac{1}{n^x}$ converges when $x\gt 1$, we can use these series for comparison. $\endgroup$ – André Nicolas Dec 25 '14 at 0:58
  • $\begingroup$ Thanks @AndréNicolas. So does it mean that I have to check the convergence of the series that I used to compare with other convergence test? Is there any other handy way to do it? $\endgroup$ – Tommy_Smith Dec 25 '14 at 1:03
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    $\begingroup$ We gather a collection of basic series that we show converge (or diverge). After that, most convergence/divergence questions, at least for series with positive terms, can be settled by Comparison or Limit Comparison. $\endgroup$ – André Nicolas Dec 25 '14 at 1:06
  • $\begingroup$ Can you please give me some example of basic series? I just know Harmonic series is obviously diverges. $\endgroup$ – Tommy_Smith Dec 25 '14 at 1:10
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    $\begingroup$ The $p$-series never converge to $0$. If $p\le 1$ they diverge, and if $p\gt 0$ they converge. A nice formula for what they converge to is only known in rare cases, most importantly when $p$ is an even positive integer. $\endgroup$ – André Nicolas Dec 25 '14 at 1:55
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When discussing series, avoid saying "series converges to ..."; this kind of statement is almost always misguided.

When discussing sequences, we talk about what their terms converge to. For example, $1/\sqrt{n}$ converges to $0$ as $n\to\infty$.

When discussing series, we could still think about what happens to individual terms, but this is not the main thing: the convergence of a series is a matter of their partial sums. The series $\sum_{n=1}^\infty 1/2^n$ converges, but it would be wrong to say that it "converges to $0$". Rather, the sequence of its terms $1/2^n$ converges to zero. The series itself converges and has sum equal to $1$.

In your examples: the sequence of terms $1/n^x$ converges to $0$ for any $x>0$.

But the series $\sum_{n=1}^\infty 1/n^x$ converges only when $x>1$. And its sum is never $0$; it's some positive number which we don't necessarily know or perhaps even care about.

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