On decreasing sequences $(a_n)_n$ such that $\sum_{n=1}^{+\infty} a_n$ diverges

Question

As an exercise for my math class, I have to find the widest possible choices of sequences $(a_n)_n$ such that $(a_n)_n$ is a nonnegative decreasing and the series $$\sum_{n=1}^{+\infty} a_n$$ diverges.

I was thinking about $a_n=\frac{1}{n}$, so that $$\sum_{n=1}^{+\infty} \frac{1}{n}$$ is the harmonic series which diverges.

Could someone please help me with more exmples?

Thank you in advance!

What exactly do you mean by "widest possible choices of sequences"? — 5xum, Oct 21 '21 at 07:34
@AlannRosas nothing more than the fact that $a_n$ has to be a decreasing sequence (I guess in the next class some we will add some more binding conditions, but that's all so far). — C. Bishop, Oct 21 '21 at 07:34
Maybe it's a requirement the terms of the sequence to be non-negative, otherwise it's too easy. — peter.petrov, Oct 21 '21 at 07:34
Note that for every diverging series $\sum a_n$ there is a “more slowly” diverging series $\sum b_n$ with $b_n/a_n \to 0$, see for example https://math.stackexchange.com/q/452053/42969 — Martin R, Oct 21 '21 at 07:35
Typical examples are $\sum \frac{1}{n \ln(n)}$,$\sum \frac{1}{n \ln (n) \ln(\ln(n))}$, ... — Martin R, Oct 21 '21 at 07:37
@User1010 What does "as many as possible" mean, mathematically? — 5xum, Oct 21 '21 at 07:39
@5xum "What does "as many as possible" mean, mathematically?" Maybe it means the limit as $n$ goes to infinity. — peter.petrov, Oct 21 '21 at 13:15

peter.petrov · Accepted Answer · 2021-10-21T13:11:41.350

0

$(1)$ Take for example

$$\sum_{n=1}^{+\infty} \frac{1}{n^\alpha}$$

where $\alpha$ is a fixed number in the interval $(0,1)$

Give different values for $\alpha$ and you will get different divergent series.

$(2)$ More examples: $$\sum_{n=1}^{+\infty} \frac{1}{n \cdot \ln(n)}$$

$$\sum_{n=1}^{+\infty} \frac{1}{\ln(n)}$$

$$\sum_{n=1}^{+\infty} \frac{1}{\ln^2(n)}$$

All these are divergent.

$(3)$ In general if $f(n)$ is a strictly increasing function which grows more slowly than $g(n) = n$ then

$$\sum_{n=1}^{+\infty} \frac{1}{f(n)}$$ is also divergent.

$(4)$ Other interesting examples would be:

$$\sum_{n=1}^{+\infty} {\sin(n)}$$

$$\sum_{n=1}^{+\infty} {\cos(n)}$$

They are also divergent. But their terms are not all non-negative.

edited Oct 21 '21 at 13:11

answered Oct 21 '21 at 07:36

peter.petrov

12,568

Something different from $sum\frac{1}{n}$ and $\sum\frac{1}{n^{\alpha}}$? – C. Bishop Oct 21 '21 at 07:57
I added a few more examples. – peter.petrov Oct 21 '21 at 08:28
1

$\sin(n)$ is neither non-negative nor decreasing. – Martin R Oct 21 '21 at 08:34
@MartinR I know. I didn't say it was. – peter.petrov Oct 21 '21 at 08:34
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But that is what the question asks for. – Martin R Oct 21 '21 at 08:35
@MartinR It asks for that after I mentioned it, and the OP changed the question. – peter.petrov Oct 21 '21 at 08:36
@MartinR I don't claim that. Maybe you misread something. – peter.petrov Oct 21 '21 at 13:06
Indeed, I misread. – Martin R Oct 21 '21 at 13:08
@MartinR I put a "But" there and some highlighting to avoid others misreading. – peter.petrov Oct 21 '21 at 13:12

On decreasing sequences $(a_n)_n$ such that $\sum_{n=1}^{+\infty} a_n$ diverges

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