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When examining a p-series (see below), any series where $p > 1$ is considered to converge. However, the stated series with $p = 1$ diverges. The only explanation I've found thus far states that the reason for this is that the series where $p = 1$ "does not tend to $0$ quickly enough". How is it determined that the series where $p = 1$ does not tend toward $0$ quickly enough to warrant convergence?

$\sum_{n=1}^\infty {1\over n^p}$

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    Look for convergence criterions of series $\sum_{n=1}^\infty a_n.$ And apply them to your cases $a_n=n^p$.2012-12-04

2 Answers 2

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You have it backwards. We know that $\left\langle\frac1n:n\in\Bbb Z^+\right\rangle$ doesn’t tend to $0$ fast enough for convergence of the harmonic series because we can prove that the harmonic series doesn’t converge. In other words, the non-convergence of the harmonic series comes first; the statement

$\left\langle\frac1n:n\in\Bbb Z^+\right\rangle$ doesn’t tend to $0$ fast enough

is just an intuitive explanation of that non-convergence.

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$1 + \frac12 + \frac13 + \frac14 + \cdots \ge 1 + \frac12 + \underbrace{\frac14 + \frac14}_{\text{$2$ terms}} + \underbrace{\frac18 + \cdots + \frac18}_{\text{$4$ terms}}+\cdots \ge 1 + \frac12 + \frac12 + \frac12 + \cdots$