Is the Example for Series in Wikipedia Incorrect?

In summary, The article says that when n is really large, the bottom gets really big and the whole fraction would head to zero. However, this is not true. A "series", a "sum of a series", a "sequence", and a "limit of a sequence" are all very different things.
  • #1
Weather Freak
40
0
Hey folks,

I'm currently studying sequences and the like in Calc 2, and I went to Wikipedia for another explanation about them. The example given in the article http://en.wikipedia.org/wiki/Series_%28mathematics%29" [Broken] seems to be incorrect to me.

The example is this:

[itex]\sum _{n=0}^{\infty }{2}^{-n} = 2[/itex]

I was thinking it's equal to zero though, since when n is really large, then the bottom gets really big so the whole fraction would head to zero.

Am I wrong or is the author wrong?
 
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  • #2
This is a series, and not a sequence.
 
  • #3
Yeah, that's what I meant to say o:) ... but should that change the answer?
 
  • #4
Well, yes. A "series", a "sum of a series", a "sequence", and a "limit of a sequence" are all very different things.
 
  • #5
You're thinking of [itex]\lim_{n \to \infty} 2^{-n}[/itex], which is zero. However [itex]\sum 2^{-n} = 1 + \frac{1}{2} + \frac{1}{4} + \cdots[/itex] isn't zero.
 
  • #6
Yes, that answer is correct. [tex]1 + \frac{1}{2} + \frac{1}{4} + \cdots[/tex] does indeed equal 2.

I probably solve simple series like this in a unique way, but I tend to imagine it in the number base 2 (binary). This would essentially be 1.11111111 repeating. This is like our 9.9999 repeating = 10, only that in binary is 2.
 
  • #7
Weather Freak said:
Am I wrong or is the author wrong?
Uhmmm, I am sorry to tell you this, but you are wrong, not the author... :tongue2:
This is a geometric series with the first term 1, and the common ratio r = 1 / 2.
So apply the formula to find the sum of the first n terms of a geometric series, we have:
[tex]S_n = a_1 \frac{1 - r ^ n}{1 - r}[/tex]
Now r = 1 / 2. So |r| < 1, that means:
[tex]\lim_{n \rightarrow \infty} r ^ n = 0[/tex]
Now let n increase without bound to get the sum:
[tex]\sum_{n = 0} ^ {\infty} 2 ^ {-n} = \lim_{n \rightarrow \infty} S_n = \lim_{n \rightarrow \infty} a_1 \frac{1 - r ^ n}{1 - r} = \frac{a_1}{1 - r} = \frac{1}{1 - \frac{1}{2}} = 2[/tex].
Can you get this? :)
-------------
@ KingNothing: Have you leant geometric series? We don't need to complicate the problem in binary, though. Just my $0.02.
 
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  • #8
Doh! I get it now! Thanks folks... and I apologize for my confusion :(... it's all a little complicated when you first learn it.
 

1. What is an "Incorrect Wikipedia Article"?

An "Incorrect Wikipedia Article" is an article on the online encyclopedia, Wikipedia, that contains incorrect or false information.

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