## ParthKohli 3 years ago Notation of a sequence: What do we mean by $$n \ge 1$$ in $$\{ 2^n\}_{n \ge 1}$$? Is it all the terms of the sequence only have $$n = 1$$ or more? So basically $$n$$ iterates to the positive numbers.

1. ParthKohli

So, the sequence is $$\{2,4,8,16\cdots \}$$.

2. ParthKohli

Am I correct?

3. ParthKohli

@agentx5 I need a genius for this.

4. agentx5

Powers of two, while n is greater than or equal to 1. Yep! You're good

5. agentx5

Infinite sequence in fact, right? (i.e.: what's the limit as n goes to infinity for 2$$^n$$ ? ) ;-)

6. ParthKohli

Ah! Great stuff! $$\{ 1,2,3,4,5,6,7\} = \{n \} _{7 \ge n \ge 1}$$

7. ParthKohli

It never reaches anywhere, so it's infinity. That was an oral question :P

8. agentx5

This is how you typically see these though, as series (sums of the terms in a sequence) $$\sum_{n \ge 1}^{\infty} 2^n$$ = $$\infty$$ or "Diverges" as they say

9. ParthKohli

|dw:1342971644614:dw|

10. ParthKohli

But what if $$n$$ is negative?

11. agentx5

n isn't negative. or zero

12. agentx5

It starts at 1.

13. ParthKohli

Oh, okay.

14. ParthKohli

What if we have negatives included?

15. agentx5

$$\{n \} _{7 \ge n \ge 1}$$ <--- this however has an upper boundary

16. agentx5

2^7 = 128

17. agentx5

so... {2,4,8,16,32,64,128}

18. ParthKohli

The limit is 0 if we have $$\{ 2^n\}_{n \le 0}$$ right?

19. ParthKohli

I believe that the sequence $$\{2,4,8,16,32,64,128 \}$$ doesn't have a limit.

20. agentx5

Well for one thing you've got the wrong graph here, what happens when you raise to a negative power?

21. ParthKohli

Exactly, we have it getting closer to 0.

22. agentx5

|dw:1342971841230:dw|

23. ParthKohli

Starts at 1 and keeps getting closer to 0.

24. ParthKohli

Hmm.

25. ParthKohli

Oops. I posted the graph for $$x^2$$.

26. ParthKohli

$\lim _{x \to -\infty} 2^x = 0$ Good enough?

27. agentx5

Yep sure does, as you approach negative infinity $\huge \lim_{x \rightarrow -\infty} 2^x = 2^{-\infty} = (\frac{1}{2^{\infty}}) = \frac{1}{\infty} = 0$

28. agentx5

See how the Algebra keeps coming back?

29. ParthKohli

It does =)

30. ParthKohli

I believe that I'd owe you a lot when I get into MIT :)

31. agentx5

I'm used to tutoring students with learning disabilities :-) That's my part-time job on campus. Maybe you can help me find a job at some point in the future, ya never know ^_^

32. ParthKohli

Heh. You're a nice guy!

33. ParthKohli

And nice guys don't finish last ;)