Why does the harmonic series diverge?

Question

Since the convergence of $\sum_{n=0}^\infty x_n$ implies $\lim_{x\to\infty} x_n$ = 0, why can't we say that $\lim_{x\to\infty} (1/x_n)$ = 0 implies the convergence of $\sum_{n=0}^\infty (1/x_n)$? If so, tbe harmonic series should converge, but it doesn’t. I want to know what went wrong.

Answer 1

You mean since:

$\sum_{n=1}^{\infty} x_n = x< \infty \Rightarrow \lim_{n\to\infty} x_n=0$

Why not:

$\lim_{n\to\infty} x_n=0 \Rightarrow \sum_{n=1}^{\infty} x_n = x< \infty $

Well first of all as said before the statement $A \Rightarrow B$ does not imply $B \Rightarrow A$. Of course sometimes the reverse also holds, and sometimes it doesn't. One has to prove equivalence or disprove it.

A simple example is sufficient as a proof that the reverse does not hold

$S = 1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+...$

$S > 1/2+1/2+1/4+1/4+1/8+1/8+1/8+1/8+... = 1/2+1/2+1/2+1/2+... = \infty$

Hence also $S=\infty$

So we can see that even though the term $1/n$ converges to $0$ the series of the sum doesn't.

So yeah here is the proof. But to me it wasn't really enough. I did find it kind of counterintuitive. It was strange to me how that by adding smaller and smaller 'pieces' (that eventually should amount to nothing) could explode to infinity. So how by adding 'almost nothing' can you get infinity?

So I was still left asking the question why and how. The proof kind of gives you a hint. As you can see in the proof above, those 'smaller pieces' are 'not that small' since by adding some of them up you get more than $1/2$ as many times as you want.

So I did a little reading later on and got my answer. It has to do with the speed of convergence. In this case, $1/n$ is 'slowly' converging to $0$.

In general $\sum_{n=1}^{\infty}1/n^p$ converges if $p>1$, because only then does $1/n^p$ converge 'fast enough' to $0$.

I encourage you to read further on these topics.

Answer 2

A implies B doesn't mean that B implies A

Answer 3

The issue in your reasoning comes from the direction of implication, and the mathmetical meaning of "if ... then ..." (and its various forms). In everyday use, "if A then B" may mean that A and B are either both true or both false. However, in mathematics the same statement means that if A is true, then B is true; if A is false, then B may be either true or false.

This has the effect that "if A then B" and "if B then A" have two very different meanings. For example, "if X is a square, then X is a shape" is clearly true, but "if X is a shape, then X is a square" is not.

In your case, it is true that for a series $[a_i]$, if the sum of the series converges, then the limit of its terms is zero. However, it is not true in general that if a series' limiting term is zero, then its sum converges.