Help Finding Whether A Sequence Converges

[FaraDazed]

Homework Statement

Q1 Are the following sequences divergent or convergent as n tends to infinity.

A: $\frac{5n+2}{n-1}$

B: $tan^{-1}(n)$

Homework Equations

The Attempt at a Solution

Really not sure how to show this mathematically, or even if what I have done is correct.

Part A:
$$\frac{5n+2}{n-1}=12,\frac{17}{2},\frac{22}{3},\frac{27}{4}...\\$$
So it looks as tho it converges to 0 as n tends to infinity.

Part B:
$$tan^{-1}(n) = \frac{\pi}{4},1.107,1.25...$$
From messing on the calculator I can see that it tends to pi/2 as n tends to infinity but don't know how to show it mathmatically.

I would really appreciate any feedback or advice :)

 

[Ray Vickson]

Homework Statement

Q1 Are the following sequences divergent or convergent as n tends to infinity.

A: $\frac{5n+2}{n-1}$

B: $tan^{-1}(n)$

Homework Equations

The Attempt at a Solution

Really not sure how to show this mathematically, or even if what I have done is correct.

Part A:
$$\frac{5n+2}{n-1}=12,\frac{17}{2},\frac{22}{3},\frac{27}{4}...\\$$
So it looks as tho it converges to 0 as n tends to infinity.

Part B:
$$tan^{-1}(n) = \frac{\pi}{4},1.107,1.25...$$
From messing on the calculator I can see that it tends to pi/2 as n tends to infinity but don't know how to show it mathmatically.

I would really appreciate any feedback or advice :)

You have three very similar threads, and in all of them you have shown no effort whatsoever to deal with the problems. Apply the concepts you have learned in class (or should have learned). Is there something about the material you do not understand? If so, tell us what it is.

 

[FaraDazed]

You have three very similar threads, and in all of them you have shown no effort whatsoever to deal with the problems. Apply the concepts you have learned in class (or should have learned). Is there something about the material you do not understand? If so, tell us what it is.

There are three threads as they were in one but was told to split them up.

I am finding the class very difficult, I have given the problems my best effort. I would rather know what I have done wrong so I can see how it should be set out.

I am not looking for someone to give me the answer. Just some guidance one what I have done wrong and how to set the problem out properly.

 

[haruspex]

What do you think $\frac{5n+2}{n-1}$ will tend to for very large n?

 

[FaraDazed]

OK I think I have made some progress on my understanding of what the question is actually after.

Does this make sense?

A:
$$\frac{5n+2}{n-1}$$
With this one to numerator will always be bigger, therefore the fraction will keep getting bigger so therefore it diverges to infinity?

Part B is still confusing me as to why it tends to pi/2 though, is it simply because the tangent function works between 0 and 90° and therefore the limit of the function is 90° which is pi/2?

 

[Mark44]

OK I think I have made some progress on my understanding of what the question is actually after.

Does this make sense?

A:
$$\frac{5n+2}{n-1}$$
With this one to numerator will always be bigger, therefore the fraction will keep getting bigger so therefore it diverges to infinity?

No. Notice that both the numerator and denominator get large as n gets large. The numerator will always be larger, but the whole fraction actually converges to a specific number.

Part B is still confusing me as to why it tends to pi/2 though, is it simply because the tangent function works between 0 and 90° and therefore the limit of the function is 90° which is pi/2?

This is pretty close.
If we let y = tan^-1(n), then n = tan(y).
What happens to tan(y) as y approaches $\pi/2$ from the left (i.e., for numbers smaller than $\pi/2$?

 

[FaraDazed]

No. Notice that both the numerator and denominator get large as n gets large. The numerator will always be larger, but the whole fraction actually converges to a specific number.
This is pretty close.
If we let y = tan^-1(n), then n = tan(y).
What happens to tan(y) as y approaches $\pi/2$ from the left (i.e., for numbers smaller than $\pi/2$?

OK thanks, yeah I see that now. From messing on the calculator I can see it converges to 5. I just really don't get how the hell that is shown mathematically though.

Also for the other one, I can see it gets at least as big as 10381.2, not sure if it gets bigger due to a limit of the number of keys allowed to be entered on my calculator.

 

[Mark44]

$$ \frac{5n + 2}{n - 1} = \frac{n(5 + 2/n)}{n(1 - 1/n)} = \frac n n * \frac{5 + 2/n}{1 - 1/n}$$

 

[FaraDazed]

$$ \frac{5n + 2}{n - 1} = \frac{n(5 + 2/n)}{n(1 - 1/n)} = \frac n n * \frac{5 + 2/n}{1 - 1/n}$$

But still from the third bit (the below), how does that show that is will converge to 5? (Sorry, I really am confused by the whole topic at the moment).

$$\frac n n * \frac{5 + 2/n}{1 - 1/n}$$

 

[Office_Shredder]

He meant to write $\frac{n}{n}$ not $\frac{5}{5}$.

Once you are left with
$$\frac{ 5+2/n}{1-1/n}$$
as n goes to infinity, what happens to 2/n and 1/n?

 

[Mark44]

That should be n/n. I'll fix it.

 

[FaraDazed]

He meant to write $\frac{n}{n}$ not $\frac{5}{5}$.

Once you are left with
$$\frac{ 5+2/n}{1-1/n}$$
as n goes to infinity, what happens to 2/n and 1/n?

Ah, I see. the 2/n and 1/n go to zero and then are left with 5/1 . Thanks.