Prove the series are not Cauchy using the Cauchy criterion with epsilon

Aug 2016
38
2
Romania
$b_n=\frac{x}{1}+\frac{x+1}{3}+...+\frac{x+n}{2n+1},x>1$
$c_n=\frac{1}{\sqrt1}+\frac{1}{\sqrt2}+...+\frac{1}{\sqrt n},n≥1$
I'm having troubles with the Cauchy criterion and I do not understand how to apply it when the series are not convergent. These exercises are telling me to use the Cauchy criterion
$$|b_{n+p}−b_{n}|<ε$$
.
Is the absolute value inside always positive? I've learned in school that for every negative value inside the absolute value symbol it changes to a positive value. Ex: |−9| = 9 or |−2x − 5| = 2x + 5.

However, since these series are not convergent, I need to prove that by using the criterion with epsilon. How do you apply that? If you apply the criterion, can you reduce terms if they are inside the absolute value symbol?
 
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Aug 2016
38
2
Romania
$$\sum_{n=1}^\infty \frac{n+1}{3n+2}=$$
Another exercise which is not cauchy and diverges to $\frac{1}{3}$
 
Aug 2016
38
2
Romania
If you use the Cauchy criterion formula, that means you reduce some terms inside the absolute value symbol |b(n+p)-b(n)|?
I really need help for these exercises. They are extremely difficult and I do not understand how to solve them.
 
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SDK

Sep 2016
689
463
USA
A (real) sequence, $\{a_n\}$ is Cauchy if for any $\epsilon > 0$, you can find an integer, $N$ so that if $a_n, a_m$ are any two terms with $n,m > N$, then $|a_n - a_m| < \epsilon$. Roughly speaking, a sequence is Cauchy if the "tail" terms are eventually isolated together into an interval of arbitrarily small diameter.

The Cauchy criteria says that a sequence converges if and only if it is a Cauchy sequence. This should match with the intuition above. If a sequence eventually stops varying, then it must converge to a constant.

I think what matters for you is the the "only if" part which says that if a sequence isn't Cauchy (i.e. if its tail is not eventually bound by arbitrarily small intervals), then your sequence can't converge. This gives a natural way to determine when a sequence doesn't converge by instead showing it is not Cauchy. This is extremely helpful since it is often much harder to show that a limit does not exist, than to prove it does.



$$\sum_{n=1}^\infty \frac{n+1}{3n+2}=$$
Another exercise which is not cauchy and diverges to $\frac{1}{3}$
What does this mean? How can a series diverge to $\frac{1}{3}$? Do you know what it means for a series to diverge?