# Why does the collection $\sum_{n=1}^\infty\frac1n$ not merge?

Can a person offer a straightforward description regarding why the

$$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$

does not merge, on the various other hand it expands really gradually?

I would certainly favor a conveniently understandable description as opposed to a strenuous evidence consistently located in undergraduate books.

This is not as excellent a solution as AgCl's, however individuals might locate it intriguing.

If you're made use of to calculus after that you could see that the amount $$ 1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n}$$ is really near the indispensable from $1$ to $n$ of $\frac{1}{x}$. This precise indispensable is ln (n), so you need to anticipate $1+\ frac 1 2 +\ frac 1 3 +\ dots npls \ frac 1 n $ to grow like $ \ ln (n) $.

Although this argument can be made strenuous, it's still unfulfilling due to the fact that it relies on the reality that the by-product of $\ln(x)$ is $\frac{1}{x}$, which is possibly harder than the initial inquiry. However it does highlight an excellent basic heuristic for promptly establishing just how amounts act if you currently recognize calculus.

Let's team the terms as adheres to :

Group $1$ : $\displaystyle\frac11\qquad$ ($1$ term)

Group $2$ : $\displaystyle\frac12+\frac13\qquad$ ($2$ terms)

Group $3$ : $\displaystyle\frac14+\frac15+\frac16+\frac17\qquad$ ($4$ terms)

Group $4$ : $\displaystyle\frac18+\frac19+\cdots+\frac1{15}\qquad$ ($8$ terms)

$\quad\vdots$

In basic, team $n$ has $2^{n-1}$ terms. Yet additionally, notification that the tiniest component in team $n$ is bigger than $\dfrac1{2^n}$. As an example all components in team $2$ are bigger than $\dfrac1{2^2}$. So the amount of the terms in each team is bigger than $2^{n-1} \cdot \dfrac1{2^n} = \dfrac1{2}$. Given that there are definitely several teams, and also the amount in each team is bigger than $\dfrac1{2}$, it adheres to that the complete amount is boundless.

This evidence is usually credited to Nicole Oresme.

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