Prove that the sequence defined by partial sums of the Harmonic series
$$\left\{S_n\right\}^{\infty}_{n=1} =\left\{\sum_{k=1}^{n} \frac{1}{k}\right\}^{\infty}_{n=1} $$
is not a Cauchy sequence.
I have a theorem that states
Theorem: A sequence is Cauchy if and only if it is convergent.
I first need help in understanding the proof that the Harmonic series is divergent. I somewhat understand that since its a sum and not a sequence that the sum will obviously go to infinity.I however get confused with the "sequence defined by partial sums of the Harmonic series" part.
second I was wondering that after proving that the sequence is indeed divergent that I can use the theorem to say its not cauchy. How would I go about wording that?
Best Answer
It might be more instructive not to use the theorem at all.
You can prove directly that $S_n=\sum^n_{k=1}\frac{1}{k}$ is not Cauchy: if $n>m,$ we have $S_n-S_m=\frac{1}{m+1} + \frac{1}{m+2} +...+ \frac{1}{n} > \frac{n - m}{n} = 1 - m/n.$ Now, let $\epsilon=1/2.$ Then, if $n>2m,\ S_n-S_m> 1/2$ and so $(S_n)$ is not Cauchy.