ParthKohli
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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So, the sequence is \(\{2,4,8,16\cdots \}\).
ParthKohli
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Am I correct?
ParthKohli
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@agentx5 I need a genius for this.
agentx5
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Powers of two, while n is greater than or equal to 1. Yep! You're good
agentx5
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Infinite sequence in fact, right? (i.e.: what's the limit as n goes to infinity for 2\(^n\) ? ) ;-)
ParthKohli
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Ah! Great stuff!
\(\{ 1,2,3,4,5,6,7\} = \{n \} _{7 \ge n \ge 1}\)
ParthKohli
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It never reaches anywhere, so it's infinity.
That was an oral question :P
agentx5
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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
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|dw:1342971644614:dw|
ParthKohli
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But what if \(n\) is negative?
agentx5
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n isn't negative. or zero
agentx5
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It starts at 1.
ParthKohli
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Oh, okay.
ParthKohli
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What if we have negatives included?
agentx5
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\(\{n \} _{7 \ge n \ge 1}\) <--- this however has an upper boundary
agentx5
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2^7 = 128
agentx5
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so...
{2,4,8,16,32,64,128}
ParthKohli
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The limit is 0 if we have \(\{ 2^n\}_{n \le 0}\) right?
ParthKohli
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I believe that the sequence \(\{2,4,8,16,32,64,128 \}\) doesn't have a limit.
agentx5
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Well for one thing you've got the wrong graph here, what happens when you raise to a negative power?
ParthKohli
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Exactly, we have it getting closer to 0.
agentx5
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|dw:1342971841230:dw|
ParthKohli
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Starts at 1 and keeps getting closer to 0.
ParthKohli
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Hmm.
ParthKohli
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Oops. I posted the graph for \(x^2\).
ParthKohli
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\[\lim _{x \to -\infty} 2^x = 0\]
Good enough?
agentx5
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Yep sure does, as you approach negative infinity
\[\huge \lim_{x \rightarrow -\infty} 2^x = 2^{-\infty} = (\frac{1}{2^{\infty}}) = \frac{1}{\infty} = 0\]
agentx5
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See how the Algebra keeps coming back?
ParthKohli
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It does =)
ParthKohli
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I believe that I'd owe you a lot when I get into MIT :)
agentx5
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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
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Heh. You're a nice guy!
ParthKohli
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And nice guys don't finish last ;)