Show if the sequence converges

[vipertongn]

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?

 

[LCKurtz]

It is really trivial to show that sequence is decreasing. Have you tried?

 

[vipertongn]

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)

 

[LCKurtz]

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 a_n+1 < a_n with reversible steps? Try putting the 2's on one side by themselves.

 

[vipertongn]

That's good advice. What happens if you work on the inequality a_n+1 < a_n 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. Let's 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

 

[LCKurtz]

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. Let's 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.

 

[vipertongn]

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?

 

[LCKurtz]

how do i simplify 2ⁿ⁺¹/1ⁿ? does it become 1 or something?

$$\frac{2^{n+1}}{2^n}\neq\frac{2^{n+1}}{1^n}$$