Show if the sequence converges

  • Thread starter vipertongn
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Homework Statement


showing it is increasing with an upper bound or decreasing
with a lower bound

n/(2^n)

Homework Equations



if an >an+1 its underbound
an< an+1 then its upperbound

The Attempt at a Solution


I tried first by finding out the sequence:
a1=1/2
a2=1/2
a3=3/8
a4=1/4
a5=5/32

I'm assuming its decreasing, but I'm not sure if this is monotonic at all. Considering how a1=a2 and then a2>a3 and a3>a4 then a4>a5. I think it's underbound since an>an+1 but the first part threw me off since a1=a2. Someone clarify for me?
 

Answers and Replies

  • #2
LCKurtz
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It is really trivial to show that sequence is decreasing. Have you tried?
 
  • #3
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Yea I can tell that the sequence is decreasing just by looking at that. I was taught to try (an+1)-an see if that's greater than 0 or not, but its kinda confusing with this equation. then there is also taking the derivative but it also doesn't help much because that ends up being just 1/2^n-1/2^n(log2)
 
  • #4
LCKurtz
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Yea I can tell that the sequence is decreasing just by looking at that. I was taught to try (an+1)-an see if that's greater than 0 or not, but its kinda confusing with this equation.

That's good advice. What happens if you work on the inequality an+1 < an with reversible steps? Try putting the 2's on one side by themselves.
 
  • #5
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That's good advice. What happens if you work on the inequality an+1 < an with reversible steps? Try putting the 2's on one side by themselves.

Ummm... do you think you could give me an example as to how to go about this. I have other problems and this one seems the most simple to actually work with. >.<;; my professor doesn't give good examples and that's usually how I work through my math. Lets see if I can get this right

so n+1/(2n+1< n/2n
That turns into n+1/n < 2n+1/2n If i subtract that...
so...
n+1/n-2n+1/2n<0 so that shows that its decreasing and underbounded (meaning its bounded under something right?

but wait if i do it the other way... isn't 0>2n+1/2n-n+1/n
 
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  • #6
LCKurtz
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Ummm... do you think you could give me an example as to how to go about this. I have other problems and this one seems the most simple to actually work with. >.<;; my professor doesn't give good examples and that's usually how I work through my math. Lets see if I can get this right

so n+1/(2n+1< n/2n
That turns into n+1/n < 2n+1/2n If i subtract that...

You need parentheses around the n+1 on the left don't you? Don't subtract anything. Just look at what you have. Simplify the right side. Compare it to the left.
 
  • #7
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You need parentheses around the n+1 on the left don't you? Don't subtract anything. Just look at what you have. Simplify the right side. Compare it to the left.

how do i simplify 2n+1/1n? does it become 1 or something?
 
  • #8
LCKurtz
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how do i simplify 2n+1/1n? does it become 1 or something?

[tex]\frac{2^{n+1}}{2^n}\neq\frac{2^{n+1}}{1^n}[/tex]
 

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