Is the Example for Series in Wikipedia Incorrect?

[Weather Freak]

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" seems to be incorrect to me.

The example is this:

\sum _{n=0}^{\infty }{2}^{-n} = 2

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?

 

[devious_]

This is a series, and not a sequence.

 

[Weather Freak]

Yeah, that's what I meant to say ... but should that change the answer?

 

[Hurkyl]

Well, yes. A "series", a "sum of a series", a "sequence", and a "limit of a sequence" are all very different things.

 

[devious_]

You're thinking of \lim_{n \to \infty} 2^{-n}, which is zero. However \sum 2^{-n} = 1 + \frac{1}{2} + \frac{1}{4} + \cdots isn't zero.

 

[KingNothing]

Yes, that answer is correct. 1 + \frac{1}{2} + \frac{1}{4} + \cdots 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.

 

[VietDao29]

Am I wrong or is the author wrong?

Uhmmm, I am sorry to tell you this, but you are wrong, not the author...
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:
S_n = a_1 \frac{1 - r ^ n}{1 - r}
Now r = 1 / 2. So |r| < 1, that means:
\lim_{n \rightarrow \infty} r ^ n = 0
Now let n increase without bound to get the sum:
\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.
Can you get this? :)
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@ KingNothing: Have you leant geometric series? We don't need to complicate the problem in binary, though. Just my $0.02.

 

[Weather Freak]

Doh! I get it now! Thanks folks... and I apologize for my confusion :(... it's all a little complicated when you first learn it.