I have to show if one of these are cauchy sequences:
$Sequence\ 1:\ x_n=\Sigma_{t=1}^{t=n}\ 1/t$
I have done some work and I believe this is not a cauchy sequence for the fact that $x_n>x_{n+1}$ and it is not bounded.
While the second one
$Sequence\ 2:x_n=\Sigma_{t=1}^{t=n}\ 1/t^2$
Although I believe it's not cauchy because $x_n>x_{n+1}$ and not being bounded I have made a simulation and it appears to converge to 1.65.So I'm at a loss
Thanks in advance
in R, a Sequence being a Cauchy sequence, is equivalent to the series converging. so what you need to check, is if the given sequences converge. the first sequence is the harmonic series, and you can read why it diverges here: https://www.wikiwand.com/en/Harmonic_series_(mathematics) the second sequence does converge, you can see proof here: Proving $\frac{1}{n^2}$ infinite series converges without integral test
It looks like you don't understand the definition of a Cauchy sequence. Essentially it means $|x_n-x_m|$ is arbitrarily small for large $m$ and $n$.
Sequence 1) is not Cauchy since $|x_n-x_m|$~$|ln(n)-ln(m)|$.
Sequence 2) is Cauchy since $|x_n-x_m|$~$2|\frac{1}{m}-\frac{1}{n}|$.
