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Determine whether the following sequence converges or diverges. If it converges find the limit. an=n(sin(1/n))

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Originally posted by Arachnarchist
Determine whether the following sequence converges or diverges. If it converges find the limit. an=n(sin(1/n))
We can rearrange this sequence and make a simple substitution to make it more recognizable:

a(n) = n*sin(1/n) = sin(1/n)/(1/n)

Let x = 1/n. Then we have:

a(x) = sin(x)/x

As n->inf, x->0, so we must find lim(x->0) a(x). This is easily done using L'Hopital's rule:

lim(x->0) sin(x)/x = lim(x->0) cos(x)/1 = 1

So this sequence does converge to a value of 1.

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Originally posted by Arachnarchist
Determine whether the following sequence converges or diverges. If it converges find the limit. an=n(sin(1/n))
Well the limit of the general term when n tends to infinity is
lim an = lim n*sin(1/n) = lim n/n = 1.
Since the general term doesn't tend to zero we can say that the series diverges.

Edit: Somehow I read your post as a question if the series of general term n*sin(1/n) converges. Sorry about that. Having read PB's answer I realised that I read it wrong.

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Originally posted by Arachnarchist
Determine whether the following sequence converges or diverges. If it converges find the limit. an=n(sin(1/n))
As for n -> 0

sin(1/n) is bounded, so n*sin(1/n) -> 0

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Originally posted by TheMaster37
As for n -> 0

sin(1/n) is bounded, so n*sin(1/n) -> 0
It's n*sin(1/n) not 1/n*sin(1/n).
The first time I also read this wrong though not like you

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Originally posted by TheMaster37
As for n -> 0

sin(1/n) is bounded, so n*sin(1/n) -> 0
I think he meant n -> infinity, not n -> 0. I'll go with what PBE6 said.

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Originally posted by adam warlock
Edit: Somehow I read your post as a question if the series of general term n*sin(1/n) converges. Sorry about that. Having read PB's answer I realised that I read it wrong.
Sequence v Series 🙂

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Originally posted by adam warlock
It's n*sin(1/n) not 1/n*sin(1/n).
The first time I also read this wrong though not like you
One thing you learn in time is to read carefully. It makes life so much easier at times 😉

Tohugh I occasionally muck up as well :p

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Originally posted by TheMaster37
One thing you learn in time is to read carefully. It makes life so much easier at times 😉

Tohugh I occasionally muck up as well :p
LOL
The thing is that I'm only used to calculating limits as n goes to infinity is this types of exercises. I think now that I didn't even read your post 😳. Automod was on full power.

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Originally posted by Arachnarchist
Determine whether the following sequence converges or diverges. If it converges find the limit. an=n(sin(1/n))
why "calculus"?

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Originally posted by genius
why "calculus"?
Because you have to take the limit as "n" tends toward infinity to see if the sequence converges or not. That's some straight-up calculus, yo.

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Originally posted by PBE6
Because you have to take the limit as "n" tends toward infinity to see if the sequence converges or not. That's some straight-up calculus, yo.
*Bows his head in shame as he re-learns what he thought the Calculus was*

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Originally posted by PBE6
Because you have to take the limit as "n" tends toward infinity to see if the sequence converges or not. That's some straight-up calculus, yo.
What does PBE6 mean? You want 1,000,000 Pall Bearers?🙂

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Originally posted by sonhouse
What does PBE6 mean? You want 1,000,000 Pall Bearers?🙂
Pirate Banana Evildoer #6.

http://www.hello-cthulhu.com/?date=2004-09-28

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