I've been tasked to prove that the sequence of partial sums of harmonic series diverges.
$$\lim_{n\to\infty}\;\;\sum_{i=1}^n \frac{1}{i}=\infty$$
I decided to prove this using the Monotone convergence theorem.
I managed to prove that the sequence in monotonic using induction.
However, I don't know how to prove that it is unbounded.
Here's an example how to do this for:
$X_n = n$
$$\lim_{n\to\infty} n = \infty $$
$∀ε>0,\;\; ∃N(ε)∈\mathbb{N} : ∀n > N(ε), |{X_n}| > ε$
$n > ε$
$N(ε) = \lfloor ε\rfloor + 1$
So for every epsilon we can find a number $N(ε)$ that for every number greater than this number the $n$-term of a sequence will be greater than epsilon.
However, I have no idea which $N$ with respect to epsilon should I take in the case of partial sums, because there is that $∑$ sign.
Thank you so much!
Best Answer
One can use the comparison test.
$$1 + \frac12 + \frac13 + \frac14 + \ldots > 1 + \frac12 + \frac14 + \frac14 + \ldots$$
So, in the RHS we can group the terms that add up to one. Take limits and there you go.