# Thread: Sequences, convergence and divergence.

1. ## Sequences, convergence and divergence.

Let {$\displaystyle {a_n}$} be a sequence and k be a positive integer.

a) If {$\displaystyle {a_n}$} converges to L, what are the limits of {$\displaystyle {a_{99+n}}$} and {$\displaystyle {a_{k+n}}$}? Why?

b) If {$\displaystyle {a_n}$} diverges, what is the limit of {$\displaystyle {a_{k+n}}$}? Why?

2. There's a theorem somewhere to the effect that if a sequence tends to a limit, then any subsequence of that sequence tends to the same limit. Let me go away and find it ...

Aha, found it:

http://www.proofwiki.org/wiki/Limit_of_a_Subsequence

... work in progress to expand it to make it more general, but it's there for the real number plane.

3. i read this over and over and i still have no idea how to do this question haha

4. Originally Posted by qzno
i read this over and over and i still have no idea how to do this question haha
May be the following fill be helpful.
Just think that you walk from the point $\displaystyle A$ to the point $\displaystyle B$.
And suppose that there is no other path is possible, i.e., you go on a unique path, and while walking mark your steps with numbers, step $\displaystyle 1$, step $\displaystyle 2$, and so on.
Say you have reached to $\displaystyle B$ after $\displaystyle 100$ steps.
Now, can you tell me, could you reach at $\displaystyle B$ without making steps $\displaystyle 1,2,3,4,5$ if you had chance to start from the point that you made step $\displaystyle 6$?
Or if you had enough long legs, could you jump from the place of step $\displaystyle 2$ to the place of step $\displaystyle 4$, then to the place of step $\displaystyle 6$, and so on, to reach $\displaystyle B$?

What you wrote above is exactly the same, $\displaystyle a_{n}\to L$ as $\displaystyle n\to\infty$, and what happens if I ignore the first $\displaystyle 99$ terms? I will still be able to reach to $\displaystyle L$ by running on the remaining terms of the sequence...

I think it is now more clear, let me know if not...

5. So your saying that I would get the same answer from an expression marked {$\displaystyle {a_n}$} as i would an expression {$\displaystyle {a_{99+n}}$}?

6. Originally Posted by qzno
So your saying that I would get the same answer from an expression marked {$\displaystyle {a_n}$} as i would an expression {$\displaystyle {a_{99+n}}$}?
of course...