Show that if $p>1$, $\sum\frac{1}{n^{p}}$ converges and if $p<1$ it diverges for $p\in\mathbb{R}^{+}$.
Is there any way to show another series converges or diverges and then use the Comparison Test to prove this?
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$\begingroup$Lemma 1 (Cauchy criterion). Let $(a_n)_{n=1}^\infty$ be a decreasing sequence of non-negative real numbers (so $a_n\ge0$ and $a_{n+1}\le a_n$ for all $n\ge 1$). Then the series $\sum_{n=1}^\infty a_n$ is convergent if and only is the series $$\sum_{k=0}^\infty2^ka_{2^k}=a_1+2a_2+4a_4+8a_8+\dotsb$$ is convergent.
Applying this lemma we can prove:
Proof. The sequence $(1/n^p)_{n=1}^\infty$ is non-negative and decreasing (why?), and so the Cauchy criterion applies. Thus this series is convergent if and only if $$\sum_{k=0}^\infty2^k\frac{1}{(2^k)^p}$$ is convergent. But by the laws of exponentiation we can write this as the geometric series (why?)$$\sum_{k=0}^\infty(2^{1-p})^k.$$
The geometric series $\sum_{k=0}^\infty x^k$ converges if and only if $|x|<1$ (why?). Thus the series $\sum_{n=1}^\infty1/n^p$ will converge if and only if $|2^{1-p}|<1$, which happens if and only if $p>1$ (why? Try to proving it just using the laws of exponentiation, and without using logarithms).
$\endgroup$ $\begingroup$You can show this using the integral test: $\sum \frac{1}{n^p} $ behaves as $\int_1^\infty dx/x^p$ which diverges if $p \leq 1$.
$\endgroup$ 2 $\begingroup$This is a great question.
Consider $S$
$$S = \sum_{n=1}^{\infty} \frac{1}{n^p}$$
Where $p$ is even, $p > 1$
Consider $f(z) = 1/z^p$ such that $z^p = 0 \implies z = 0$
So $z$ is a pole. By the residue theorem,
$$\sum_{n=-\infty}^{\infty} \frac{1}{n^p} = -\sum \text{sum of residues of} \space \pi\cot(\pi z) f(z)$$
Because $p$ is even,
$$\sum_{n=-\infty}^{\infty} \frac{1}{n^p} = \sum_{n=1}^{\infty} \frac{1}{n^p} + \sum_{n=-\infty}^{1} \frac{1}{n^p} = 2\sum_{n=1}^{\infty} \frac{1}{n^p}$$
The residues are:
$$\lim_{z \to 0} \frac{\pi\cot(\pi z)}{z^{p-1}} = 2\sum_{n=1}^{\infty} \frac{1}{n^p} $$
Which exists for $p > 1$, even and you see that the denominator grows too slowly if $p < 1$ which means that limit wont exist.
This however only works in my knowledge for $p$ even.
$\endgroup$ $\begingroup$There are many ways to skin this particular cat. Among methods not already mentioned, perhaps my favorite is the use of Cauchy's Condensation Test. See $\S 2.4.3$ of these notes for a statement of CCT and its application to convergence of $p$-series.
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