1. ## Sequences

I'm a newbie so didn't really know where to post this, but I hope someone here can help me!I'm just beginning to study sequences, and I'm getting nowhere with it. Can anyone please help me with these two problems:

1.)Show that the sequence (1/n^k)nEN is convergent if and only if k>=0, and that the limit is 0 for all k>0.

2.)Determine the least value of N such that n/(n^2+1)<0.0001 for all n>=N

2. Originally Posted by simplysparklers
I'm a newbie so didn't really know where to post this, but I hope someone here can help me!I'm just beginning to study sequences, and I'm getting nowhere with it. Can anyone please help me with these two problems:

1.)Show that the sequence (1/n^k)nEN is convergent if and only if k>=0, and that the limit is 0 for all k>0.

2.)Determine the least value of N such that n/(n^2+1)<0.0001 for all n>=N

$\displaystyle p \implies q$

Since $\displaystyle a_n$ is convergent it must be bounded(why?)

Now consider the three cases k < 0; k=0; k> 0

if k < 0 then k = -m for some m > 0

But then $\displaystyle \frac{1}{n^k}=\frac{1}{n^{-m}}=n^m$
But this cannot happen because the above is not bounded.

If k=0 $\displaystyle a_n=1$ for all n

if k > 0 the sequence in monotically decreasing $\displaystyle a_{n+1}< a_{n}$ for all n

This in bounded below by zero and must converge to zero.

$\displaystyle q \implies p$ is fairly straight forward. use the definition

Hint: let $\displaystyle \epsilon > 0$ Choose $\displaystyle N=\left( \frac{1}{\epsilon} \right)^{1/k}$

For #2

Clear the fraction to get $\displaystyle n=0.0001n^2+0.0001$

Good luck.

3. On part 2 though, why is it $\displaystyle n=0.0001n^2+0.0001$?and not $\displaystyle n<0.0001n^2+0.0001$?

4. Originally Posted by simplysparklers
On part 2 though, why is it $\displaystyle n=0.0001n^2+0.0001$?and not $\displaystyle n<0.0001n^2+0.0001$?
It is. It is just easier to solve the equality to get n=9999.9999 but since n needs to be a natural number we choose 10,000.

P.S we don't use the other root. (why?)

I hope this helps.

5. It does help, thank you so much!

I do have to ask though, why isn't the other root used??

& in the first question, ok, I get the first part, and taking the diferent values for k, and I understand in my head because it is bounded below by 0, it must converge to 0, but I don't get the proof? Like, what do I do with
Hint: let Choose ?Sorry about this, but could you spell it out for me please?I'm never hit such a absolute immovable wall before over a maths topic, but I am not getting sequences at all and I have this assignment due in soon Thank you so much for your time and help!

6. Originally Posted by simplysparklers
It does help, thank you so much!

I do have to ask though, why isn't the other root used??

& in the first question, ok, I get the first part, and taking the diferent values for k, and I understand in my head because it is bounded below by 0, it must converge to 0, but I don't get the proof? Like, what do I do with
Hint: let Choose ?Sorry about this, but could you spell it out for me please?I'm never hit such a absolute immovable wall before over a maths topic, but I am not getting sequences at all and I have this assignment due in soon Thank you so much for your time and help!

We wish to show that $\displaystyle |a_n-L| < \epsilon$

let $\displaystyle \epsilon > 0$ be given

Let $\displaystyle N=\left( \frac{1}{\epsilon} \right)^{1/k}$

Then for all $\displaystyle n> N$ we get...

$\displaystyle \left( \frac{1}{\epsilon}\right)^{1/k}<n$

moving some factors around we get

$\displaystyle \frac{1}{n}<(\epsilon)^{\frac{1}{k}}$

Note: the above manipulation is okay becuase n and epsilon are both positive.

Rasing both sides to the kth power we get

$\displaystyle \frac{1}{n^k}< \epsilon$ we will use this to prove what we want.

Now for all $\displaystyle n>N$ we get

$\displaystyle |a_n-L|=|\frac{1}{n^k}-0|=|\frac{1}{n^k}|<\epsilon$

QED

7. Originally Posted by TheEmptySet
It is. It is just easier to solve the equality to get n=9999.9999 but since n needs to be a natural number we choose 10,000.

P.S we don't use the other root. (why?)

I hope this helps.
Thank you so much!I get all of the first one now!
The thing I still don't get about the 2nd one,(& I'm sorry to keep bothering you on this!), is that normally you have n> [some equation with epsilon], but in his case you are just given the value of epislon, so you don't have to sub a value for epsilon into an equation to get the value of N...so where do you get the value of N?How does having n=10,000 help?

And a follow up question, you know the way it should be |a_n-L|< epsilon whenever n > N, well what if the sign is reversed?And it's |a_n-L|>= epislon for all n > N? It's in a similar question to the above one, i.e.:
Determine the least value of N such that n^2 + 2n >= 9999 for all n>N

Thank you so much for all you help!I'm stumped!

8. Originally Posted by simplysparklers
Thank you so much!I get all of the first one now!
The thing I still don't get about the 2nd one,(& I'm sorry to keep bothering you on this!), is that normally you have n> [some equation with epsilon], but in his case you are just given the value of epislon, so you don't have to sub a value for epsilon into an equation to get the value of N...so where do you get the value of N?How does having n=10,000 help?

And a follow up question, you know the way it should be |a_n-L|< epsilon whenever n > N, well what if the sign is reversed?And it's |a_n-L|>= epislon for all n > N? It's in a similar question to the above one, i.e.:
Determine the least value of N such that n^2 + 2n >= 9999 for all n>N

Thank you so much for all you help!I'm stumped!

We are trying to find the smallest natural number N such that the inequality is true

for example with the equation $\displaystyle n^2+2n \ge 9999$

if we solve this for equality (and get a fraction) we can than choose the next natural number to make the inequality hold

so we want to solve
$\displaystyle n^2+2n=9999 \iff n^2+2n-9999=0 \iff (n+101)(n-99)=0$

so we get 2 solutions n=99 or n =-101
we get rid of -101 becuase it is not a natural number i.e (positive integer)

The above inequality will hold for all $\displaystyle n \ge 99$

we can check this in the original inequality

$\displaystyle 99^2+2(99)=9801+198=9999 \ge 9999$

if you check anhy number greater than 99 it will work but any number less than 99 will not work.

I hope this clears it up.

9. It certainly does TheEmptySet!!Thank you so so much for your help!!You rock!!!