calculus - Convergence of series of the form $sum 1/n^x$

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)?

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?

Answer

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.