Does the Series \(\sum_n (\sin(n))^n\) Converge?

[Take_it_Easy]

This is not an homework, but just a couriosity that I have
I never found a easy way to solve the question of convergence of this serie

$$\sum_n (\sin(n))^n$$

any help is appreciated!

 

[MathematicalPhysicist]

look at exp(in)=cos(n)+isin(n)
(exp(in))^n=exp(in^2)=cos(n^2)+isin(n^2)=(cos(n)+isin(n))^n
Now because exp(in^2) converges also sin(n)^n converges.

I hope I haven't done a mess with the question here.

 

[nicksauce]

Based on a maple plot I have made, it appears not to converge.
Plotted is
$$f(X) = \sum_{n=1}^{n=X}\sin{n}^n$$
from X=0,100000

 

[sutupidmath]

Or, i guess we could have also reasoned this way:

SInce |sin(n)|<1 for all n positive integers. then the series

$$\sum_{n=1}^{\infty}|sin(n)|^n$$ converges, so the original series converges absolutely=> it also simply converges.

 

[Take_it_Easy]

Or, i guess we could have also reasoned this way:

SInce |sin(n)|<1 for all n positive integers. then the series

$$\sum_{n=1}^{\infty}|sin(n)|^n$$ converges, so the original series converges absolutely=> it also simply converges.

I wish we could! :)

think of
$$\sum_{n=1}^{\infty} \left( 1 - {1 \over n} \right)^n$$

 

[Take_it_Easy]

Based on a maple plot I have made, it appears not to converge.
Plotted is
$$f(X) = \sum_{n=1}^{n=X}\sin{n}^n$$
from X=0,100000

The proof I wrote actually shows it DOES NOT converge.

I can't proof it diverges to +infinity yet.

My proof is very tecnical and LONG and actually there are some points a bit difficoult that I have to clear out before to call an EXACT proof.

I need help!

 

[Take_it_Easy]

look at exp(in)=cos(n)+isin(n)
(exp(in))^n=exp(in^2)=cos(n^2)+isin(n^2)=(cos(n)+isin(n))^n
Now because exp(in^2) converges also sin(n)^n converges.

I hope I haven't done a mess with the question here.

I can't get why exp(in^2) converges.

Can you explain, please?

 

[sutupidmath]

I wish we could! :)

think of
$$\sum_{n=1}^{\infty} \left( 1 - {1 \over n} \right)^n$$

I wish too!

I wrote before i gave it a thought!

 

[v2kkim]

Please help me find this simpler limit, no summation. Probably I knew before but not now.

$$\left( 1 + {a \over n} \right)^n$$
as n goes to infinity, and a is a constant.

 

[confinement]

That is a famous limit, you can find it's value by taking a logarithm and using L'Hopital's rule.

 

[lurflurf]

That is a famous limit, you can find it's value by taking a logarithm and using L'Hopital's rule.

:(

(1+1/n)^n:=exp(log(1+a/n)/n)

log(1+a/n)/n=a{log(1+a/n)/(a/n)} -> a{1}=a

since
log(1+x)/x -> log'(1)=1

 

[Wouzzat]

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

(please excuse my english, I'm not familiar with math in english, but I hope you'll understand me anyway)

with a sum : $$S = \sum^{\infty} u_n$$, if S is convergeant, then $$\lim_{n \to \infty} u_n = 0$$.
by contraposition we can say that if $$\lim_{n \to \infty} u_n \ne 0$$ then S is not convergeant, and so it is divergeant.

In this case :
$$u_n=sin(n)^n$$,
and $$-1\le sin(n)\le1$$, comment : we can't use a strict inegality here
so $$-1\le sin(n)^n\le1$$,
$$\lim_{n \to \infty} u_n \ne 0$$,
and so S is divergeant.

 

[csprof2000]

Actually, you know, I agree with wouzzat. I think he's right... any other thoughts?

 

[sutupidmath]

[tex]\$$u_n=sin(n)^n$$,<br /> and $$-1\le sin(n)\le1$$, comment : we can't use a strict inegality here<br /> so $$-1\le sin(n)^n\le1$$,<br /> $$\lim_{n \to \infty} u_n \ne 0$$,<br /> and so S is divergeant.[/tex]

$$<br /> Isn't zero (0) also somewhere between -1 and 1? I have forgotten, pardone my bad memory!<img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f61b.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":-p" title="Stick Out Tongue :-p" data-smilie="7"data-shortname=":-p" />$$

 

[csprof2000]

I believe that what he's saying is that no matter how far you go out, it will always keep hitting '1' again every 2PI steps... '-1' every once and a while, it never really approaches zero in any meaningful way.

 

[sutupidmath]

I believe that what he's saying is that no matter how far you go out, it will always keep hitting '1' again every 2PI steps... '-1' every once and a while, it never really approaches zero in any meaningful way.

Well, that might be the case, it is just that i don't think we can jumb to that conclusion solely from what is in there, since like i said, 0 is still a possibility to be eventually the limit of that function; weird stuff happens as n-->infty. we need some other way to show that. But you are right, we can see that it does not approach 0 at any finite nr n, but like i said, that is just a sign to prove that it doesn't approach 0, it is not a proof on itself.

 

[Office_Shredder]

You don't actually go out 2Pi steps in sin(n). In fact, it's fairly easy to see that sin(n) will NEVER be 1 or -1 for n a natural number. The argument doesn't conclude what the limit of sin(n)ⁿ as n goes to infinity will be

loop's argument is wrong because e^{in^2)} has absolute value one, so summing over it doesn't converge. In fact, if you can tie in the sin series with the cosine equivalent (cos(n)^n) so that one converges if and only if the other does (not sure if you can though) then this would be a proof that it doesn't converge.

I'd give my solution, but it's super top secret and doesn't exist :p

 

[csprof2000]

Good point, Office Shredder. You're right, of course...

 

[Office_Shredder]

Probably a dead end but... the question of whether sin(n)ⁿ actually goes to 0 can changed into a question of how fast you can approximate pi by rationals (since sin(n) will be close to 1 or -1 when n is close to a multiple of pi, say k*pi, so n/k is close to pi). I distinctly remember something in my algebraic number theory course about algebraic/transcendental numbers being classified based on how quickly you could converge to them based on how large the denominator is, so I'll look that up and see if it helps with anything.

Regardless, I don't think this is the intent of how the problem was supposed to be solved

 

[Wouzzat]

Isn't zero (0) also somewhere between -1 and 1? I have forgotten, pardone my bad memory!

we can see that it does not approach 0 at any finite nr n, but like i said, that is just a sign to prove that it doesn't approach 0, it is not a proof on itself.

Hum... yes, actually I didn't proved anything except that "maybe the limit is 0, maybe it isn't".

I'm curious about the final word of that...

 

[jefswat]

I'm confused. I was taught in my calc class that if the limit of the sequence is not 0 then the series diverges. But someone raised a good point. sin(x) doesn't really have a limit as x$$\rightarrow$$ infinity. Does that series converge?

 

[dvs]

sin(n)^n doesn't converge to 0.

Hint!

 

[Office_Shredder]

I'm confused. I was taught in my calc class that if the limit of the sequence is not 0 then the series diverges. But someone raised a good point. sin(x) doesn't really have a limit as x$$\rightarrow$$ infinity. Does that series converge?

sin(x) doesn't have a limit as x goes to infinity, but sin(n)ⁿ might, since raising to the nth power greatly reduces terms less than one. For example, the sequence

1/2,-1/2, 1/2, -1/2,... doesn't converge, but 1/2, (-1/2)², (1/2)³,... = 1/2,1/4,1/8,1/16,.. does converge