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.

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- ParthKohli

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- ParthKohli

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

- ParthKohli

Am I correct?

- ParthKohli

@agentx5 I need a genius for this.

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## More answers

- anonymous

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

- anonymous

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

- ParthKohli

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

- ParthKohli

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

- anonymous

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

- ParthKohli

|dw:1342971644614:dw|

- ParthKohli

But what if \(n\) is negative?

- anonymous

n isn't negative. or zero

- anonymous

It starts at 1.

- ParthKohli

Oh, okay.

- ParthKohli

What if we have negatives included?

- anonymous

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

- anonymous

2^7 = 128

- anonymous

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

- ParthKohli

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

- ParthKohli

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

- anonymous

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

- ParthKohli

Exactly, we have it getting closer to 0.

- anonymous

|dw:1342971841230:dw|

- ParthKohli

Starts at 1 and keeps getting closer to 0.

- ParthKohli

Hmm.

- ParthKohli

Oops. I posted the graph for \(x^2\).

- ParthKohli

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

- anonymous

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

- anonymous

See how the Algebra keeps coming back?

- ParthKohli

It does =)

- ParthKohli

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

- anonymous

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 ^_^

- ParthKohli

Heh. You're a nice guy!

- ParthKohli

And nice guys don't finish last ;)

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