Use Cauchy Criterion to prove the convergence of $x_n=1+\frac{1}{ 2^2}+\cdots+\frac{1}{n^2}$

Asked Oct 24 '13 at 11:30

Active Oct 24 '13 at 12:36

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Use Cauchy Criterion to prove the convergence of $x_n=1+\frac{1}{2^2}+\frac{1}{3^2}+ \ldots +\frac{1}{n^2}$

My attempt

Take $|x_m-x_n|$, where $m>n$,

We have $|x_m-x_n|=\frac{1}{(n+1)^2}+\frac{1}{(n+2)^2}+\ldots+\frac{1}{m^2}$

Then I am unable to think up of a bound that tends to zero as $n,m\to\infty$.

Can anyone help? Thanks!

edited Jun 12 '20 at 10:38

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asked Oct 24 '13 at 11:30

Yellow Skies

9

$\frac{1}{(n+1)^2} < \frac{1}{n(n+1)} = \left(\frac1n - \frac{1}{n+1}\right)$. – Daniel Fischer Oct 24 '13 at 11:31
$|x_m-x_n|=\frac{1}{(n+1)^2}+\frac{1}{(n+2)^2}+\ldots+\frac{1}{m^2}\leq\frac{1}{n(n+1)}+\frac{1}{(n+1)(n+2)}+...\frac{1}{(m-1)m}=\frac{1}{n}-\frac{1}{m}$ I still cannot think of a way to continue – Yellow Skies Oct 24 '13 at 12:37
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That's less than $\frac1n$. So, generally, $\lvert x_n - x_m\rvert < \dfrac{1}{\min {m,n}}$. – Daniel Fischer Oct 24 '13 at 12:39
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A few related questions (you can surely find more such questions here): Does $\sum\limits_{k=1}^n 1 / k ^ 2$ converge when $n\rightarrow\infty$?, Proving $\frac{1}{n^2}$ infinite series converges without integral test, Need to prove the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges – Martin Sleziak Oct 24 '13 at 12:42
thanks daniel, sorry martin, i merely searched questions titled cauchy criterion. I will try harder next time – Yellow Skies Oct 24 '13 at 12:56

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