Increasing/ Decreasing of a Sequence

[trap101]

Determine the monotonicity and boundedness of the sequence.

1) 4n/ (4n² + 1)²2) 2ⁿ/ 4ⁿ + 1Question: I'm having a problem in knowing whether the approach I'm using is providing the right solutions.

in 1) I used the a_n+1/a_n and tried to compare their ratios. I end up with: 4n+4/ (4n² + 8n + 5)^1/2 . Now I know this will be less than 1 when I use a few "test values" such as n = 1, 2, etc. But how am I certain that the direction of the sequence won't eventually change?

If I take the derivative of the original sequence I end up with (after simplifying): 16n² - 2n + 4 = 0. In that equation I can't find any critical points so is it safe to say that the sequence is always increasing based on that logic?In 2) I did the a_n+1/a_n approach and got it was decreasing. But only was I was able to conclude that was from putting in "test values" at the end again. This is when I simplified the ratio to: 2(4ⁿ+1)/ 4ⁿ⁺¹ +1. I tried to find the derivative and get critical points but I couldn't find anything. What should I do in this case?

Thanks

 

[Bohrok]

Try finding the derivative of 4x/(4x² + 1)² again.

 

[trap101]

Try finding the derivative of 4x/(4x² + 1)² again.

My mistake, it was actually suppose to be 4x/(4x² + 1)^1/2

 

[Bohrok]

Well find the derivative of that, and what do you get?

 

[trap101]

I get 16n²-2n + 4 = 0


which when i tried to solve for roots i could not find any, even using the -b + (b²- 4ac)^1/2/2a formula.

 

[Bohrok]

I'm not sure how you're getting that; try it one more time. Remember that (f/g)' = (gf' - fg')/g²

 

[trap101]

I'm not sure how you're getting that; try it one more time. Remember that (f/g)' = (gf' - fg')/g²

Maybe I should explain how I'm getting that. So after I differentiate I get this:

4(4n²+1)^1/2 - 4n(1/2(4n²+1)^-1/2) / (4n²+1) = 0


and then I simplify everything and end up with what I got above. Then to solve for possible critical points I used the quadratic formula, which doesn't work.

 

[trap101]

Maybe I should explain how I'm getting that. So after I differentiate I get this:

4(4n²+1)^1/2 - 4n(1/2(4n²+1)^-1/2) / (4n²+1) = 0


and then I simplify everything and end up with what I got above. Then to solve for possible critical points I used the quadratic formula, which doesn't work.

Blimey! I forgot to apply the chain rule to the inside. Ok, but now I ended up with 4 = 0 as my simplified expression. What does that indicate?

 

[SammyS]

Blimey! I forgot to apply the chain rule to the inside. OK, but now I ended up with 4 = 0 as my simplified expression. What does that indicate?

That means that the equation has no solution. In this particular case it means that the derivative is not zero anywhere.

Since the function, 4x/(4x² + 1)^(1/2), is differentiable for all real numbers, what does the fact that its derivative is never zero tell you?

 

[trap101]

That means that the equation has no solution. In this particular case it means that the derivative is not zero anywhere.

Since the function, 4x/(4x² + 1)^(1/2), is differentiable for all real numbers, what does the fact that its derivative is never zero tell you?

Well when I was looking over some other questions that I'm having the similar problems with what I noticed is that it indicates that either the sequence is increasing or decreasing through out. To find out which one it does exactly, are you able to just take a couple test values and plug them into observe the behavior? i.e n= 1, 2, 3, etc. Since you know it has to go in only one direction

 

[SammyS]

Well when I was looking over some other questions that I'm having the similar problems with what I noticed is that it indicates that either the sequence is increasing or decreasing through out. To find out which one it does exactly, are you able to just take a couple test values and plug them into observe the behavior? i.e n= 1, 2, 3, etc. Since you know it has to go in only one direction

That will work, or evaluate the derivative at anyone point, since it doesn't change sign.

 

[trap101]

Thanks. One other quick question about types of sequences:

2ⁿ/ 4ⁿ+1

How do I handle this sort of sequence? I tried the a_n+1/a_n approach, but I don't think it's conclusive. When I take the derivative I end up with :

(2ⁿ)(4ⁿ)(ln 2 - ln 4) = 0

this is after setting the respective powers to e^ln.

How do I determine behavior from here?

 

[Dick]

Thanks. One other quick question about types of sequences:

2ⁿ/ 4ⁿ+1

How do I handle this sort of sequence? I tried the a_n+1/a_n approach, but I don't think it's conclusive. When I take the derivative I end up with :

(2ⁿ)(4ⁿ)(ln 2 - ln 4) = 0

this is after setting the respective powers to e^ln.

How do I determine behavior from here?

You don't have to take any derivatives at all. Just divide numerator and denominator by 2^n. The answer will depend on whether you are asking about 2^n/4^n+1 or 2^n/(4^n+1).

 

[trap101]

You don't have to take any derivatives at all. Just divide numerator and denominator by 2^n. The answer will depend on whether you are asking about 2^n/4^n+1 or 2^n/(4^n+1).

I don't fully understand what you mean by dividing by 2ⁿ. So if I divide everything by 2ⁿ I should get something of this form:

1 / 2ⁿ + (1/2ⁿ)

Now since this is a sequence I'm only concerned with values of n ≥ 1. With that being the case, the fraction will go to zero eventually...Is that the right interpretation?

 

[Dick]

I don't fully understand what you mean by dividing by 2ⁿ. So if I divide everything by 2ⁿ I should get something of this form:

1 / 2ⁿ + (1/2ⁿ)

Now since this is a sequence I'm only concerned with values of n ≥ 1. With that being the case, the fraction will go to zero eventually...Is that the right interpretation?

Yes, it goes to zero. To show it's decreasing you just have to show the denominator is increasing.