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September 4th, 2019, 02:24 PM   #1
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Cauchy series criterion

I have two exercises:
$$b_n=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+.. .+\frac{1}{n^2}$$
$$n\geq1$$
$$c_n=\frac{1}{\sqrt1}+\frac{1}{\sqrt2}+...+\frac{ 1}{\sqrt{n}}$$
$$n\geq1$$
Do I replace the terms $x_{n+p}$ and $x_n$ with:
$\displaystyle \left| \frac{1}{(n+p)^2}-\frac{1}{n^2} \right| \lt ε$
$\displaystyle \left| \frac{1}{\sqrt{n+p}}-\frac{1}{\sqrt
{n}} \right| \lt ε$
in the criterion:$\left| x_{n+p}-x_n \right| \lt ε$?
That is all I have to prove if the exercise wants me to use the convergence criterion of Cauchy?

Would it be a mistake if I wrote both series like these?
$\displaystyle b_{n+p} = \frac{1}{(n+1)^2}+\frac{1}{(n+2)^2}+...+\frac{1}{( n+p)^2}$
$\displaystyle c_{n+p} = \frac{1}{\sqrt{n+1}}+\frac{1}{\sqrt{n+2}}+...+ \frac{1}{\sqrt{n+p}}$

Last edited by skipjack; September 4th, 2019 at 04:17 PM.
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September 6th, 2019, 07:20 PM   #2
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Hint $b_n$ series converges. $c_n$ series diverges.
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