what is the general way of showing a sequence diverges and how would you manipulate this method so it works for series?

Question

How could we show the divergence of a sequence solely by using the definition of divergence ? Also how can this general method be manipulated to work for series too?

Answer 1

Based on OP's comment, here is a simple example to prove the sequence with terms $a_n = n^2$ tends to infinity by using the definition; we write: $$\lim_{n \to \infty} n^2= +\infty$$ if the following is satisfied: $$ \forall H > 0, \exists N \in \mathbb{N}: \color{blue}{n > N} \implies \color{red}{n^2 > H}$$ For an arbitrary $H>0$, pick $N$ so that $N > \sqrt{H}$; you then have that whenever $\color{blue}{n > N}$: $$a_n = \color{red}{n^2} > N^2 \color{red}{>} \sqrt{H}^2=\color{red}{H}$$ so $\color{red}{n^2 > H}$.

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How do you arrive at a good choice for $N$? You usually work backwards. You want $n^2 > H$ to hold for any $H$ and you're looking for a condition on $n$ to make this work. Since $n^2 > H \Leftrightarrow n > \sqrt{H}$ if $H$ is positive, a good choice for $N$ is anything above $\sqrt{H}$.