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

Can a person offer a straightforward description regarding why the harmonic series

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

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.

2019-05-06 01:49:53
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Answers: 2

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.

2019-05-08 11:40:53

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)


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.

2019-05-08 11:31:25