Find all real numbers such that the series converges

tamintl
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Homework Statement



Find all real numbers x ≠ −1 such that the series

Ʃn=1 (1/n) . [(x-1)/(x+1)]n

converges..


Homework Equations





The Attempt at a Solution



I really do not know where to start here. I assume you could first prove it converges sing a test but I'm really not sure.

Regards
 
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tamintl said:
I really do not know where to start here. I assume you could first prove it converges sing a test but I'm really not sure.


Well no, you're trying to find x such that the series converges, not prove that the series does converge, which it doesn't for many values of x.

If I asked you to find the radius of convergence for \sum_{n=1}^{\infty}\frac{1}{n}y^{n} would you know what I was on about? Or if you haven't done radius of convergence yet, would you be perhaps able to see which values of y would make the above sum converge?
 
Last edited:
Stimpon said:
Well no, you're trying to find x such that the series converges, not prove that the series does converge, which it doesn't for many values of x.

If I asked you to find the radius of convergence for \sum_{n=1}^{\infty}\frac{1}{n}y^{n} would you know what I was on about? Or if you haven't done radius of convergence yet, would you be perhaps able to see which values of y would make the above sum converge?

Yes, i have brief experience with the radius of convergence, however I thought that more often than not you had to use the ratio test with the radius of convergence? In that example would the Radius = ∞?

What values of y would we need for it to converge?

Regards
 
Well if you have seen the radius of convergence then you should have seen the formula r=lim_{n\rightarrow\infty}|\frac{C_{n}}{C_{n+1}}| so can you see how to use that to find the values of y?

And them presuming you can see it, do you see how to find the values of x from that?
 
Stimpon said:
Well if you have seen the radius of convergence then you should have seen the formula r=lim_{n\rightarrow\infty}|\frac{C_{n}}{C_{n+1}}| so can you use that to find the values of y?
Okay, using the formula you gave me:

lim(n→∞) (n+1)/n . yn/yn+1

= lim(n→∞) (n+1)/n . 1/y

Now what?.. I can see the (n+1)/n diverges..
 
tamintl said:
Okay, using the formula you gave me:

lim(n→∞) (n+1)/n . yn/yn+1

= lim(n→∞) (n+1)/n . 1/y

Now what?.. I can see the (n+1)/n diverges..

No sorry I should have stated what the various things were.

If you have something of the form \sum_{n=1}^{\infty}C_{n}y^{n} then the radius of convergence, r is defined by the formula I gave in the last post.

And \frac{n+1}{n} doesn't diverge.
 
Stimpon said:
No sorry I should have stated what the various things were.

If you have something of the form \sum_{n=1}^{\infty}C_{n}y^{n} then the radius of convergence, r is defined by the formula I gave in the last post.

And \frac{n+1}{n} doesn't diverge.

i don't understand. :/
 
Okay so \sum_{n=1}^{\infty}\frac{1}{n}y^{n} is of the form \sum_{n=1}^{\infty}C_{n}y^{n} where C_{n}=\frac{1}{n}

Do you see now how to find the radius of convergence for the series?
 
Stimpon said:
Okay so \sum_{n=1}^{\infty}\frac{1}{n}y^{n} is of the form \sum_{n=1}^{\infty}C_{n}y^{n} where C_{n}=\frac{1}{n}

Do you see now how to find the radius of convergence for the series?

Okay.. so using that formula we get as n→∞ lim [ (n+1)/1 . 1/y ]

but since (n+1)/1 converges to 1 then we have... 1/y < 1.. therefore y>1

How's that
 
  • #10
tamintl said:
Okay.. so using that formula we get as n→∞ lim [ (n+1)/1 . 1/y ]

but since (n+1)/1 converges to 1 then we have... 1/y < 1.. therefore y>1

How's that

No, you're not finding the limit of anything involving y, you're just finding r=\lim_{n\rightarrow\infty}\frac{C_{n}}{C_{n+1}}.

And then from there you'll have -r&lt;y&lt;r as values for which the series converges which you can use to find x in your original question.
 
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  • #11
Stimpon said:
No, you're not finding the limit of anything involving y, you're just finding r=lim_{n\rightarrow\infty}\frac{C_{n}}{C_{n+1}}.

And then from there you'll have -r&lt;y&lt;r as values for which the series converges which you can use to find x in your original question.

Wow, I'm not functioning today..

So Cn=1/n

So, r = lim (1/n) / (1/n+1)

r = lim (n+1)/n

r=1
...

therefore, -1<y<1
 
  • #12
Yes! Now that you know the range of values y=(x-1)/(x+1) can take, you should be able to immediately see how to find the range of values x can take :)
 
  • #13
Stimpon said:
Yes! Now that you know the range of values y=(x-1)/(x+1) can take, you should be able to immediately see how to find the range of values x can take :)

Thanks for your help.. I still can't see how I could use them.. Am i missing something here.. There are no values in my original question.
 
  • #14
Well you have that if you want \sum_{n=1}^{\infty}\frac{1}{n}y^{n} to converge then you must have -4&lt;y&lt;4

Then you can simply say y=\frac{x-1}{x+1} and so -4&lt;\frac{x-1}{x+1}&lt;4 and from there find the range of values x can take for the original sum in your opening post to converge.
 
  • #15
Stimpon said:
Well you have that if you want \sum_{n=1}^{\infty}\frac{1}{n}y^{n} to converge then you must have -4&lt;y&lt;4

Then you can simply say y=\frac{x-1}{x+1} and so -4&lt;\frac{x-1}{x+1}&lt;4 and from there find the range of values x can take for the original sum in your opening post to converge.

Great. But why have you put the values for r = 4 ?? I thought we agreed that the r=1

Therefore we would have -1<y<1 ?

Thanks for everything
 
  • #16
Ah sorry I was very tired and mistyped, it should indeed have been -1<y<1.
 
  • #17
Stimpon said:
Ah sorry I was very tired and mistyped, it should indeed have been -1<y<1.

great no worries.

Right so I've got it as x>0 and x>-1

hows that?
 
  • #18
Almost right.
 
  • #19
Stimpon said:
Almost right.

Would it just be x>0.. since x≠-1..

So x ε ℝ+ (ie.. positive real numbers)
 
  • #20
That's exactly right :)
 

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