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I have a clueless friend who believes that

$$ \sum_{n=1}^\infty \frac{1}{2^n} $$

doesn't equal $1$ in the 'normal arithmetical sense'. He doesn't believe that this series flat out equals $$1$$ Is he correct?

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What is the "normal arithmetical sense" in which we can interpret infinite series? –  Chris Eagle May 24 '12 at 20:40
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@Fried: $\sum \frac{1}{2^n}$ is a symbol. It denotes that least upper bound. –  Qiaochu Yuan May 24 '12 at 20:44
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@Marvis: I don't understand your reasoning. –  Qiaochu Yuan May 24 '12 at 20:44
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The point is that since there is no way to sum an infinite collection of nonzero numbers one-by-one, the meaning we ascribe to $\sum_{n=1}^\infty a_n$ is the limit of the partial sums, if that limit exists. You might not call this "the normal arithmetical sense", but then it's up to you to say what (if anything) is "the normal arithmetical sense" for such a series. –  Robert Israel May 24 '12 at 20:53
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possible duplicate of Does .99999... = 1? –  MJD May 24 '12 at 20:54

3 Answers 3

As far as I can tell, your friend is not distinguishing appropriately between a series and its sum. A series is a sequence of numbers $s_1, s_2, s_3, ...$ which one specifies by specifying $a_1 = s_1, a_n = s_n - s_{n-1}$. The notation $$\sum_{n=1}^{\infty} a_n$$

denotes the limit of the sequence $s_i$, if it exists, and is called the sum of the series. It is a number which is unique if it exists and should not be identified with the series itself.

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+1) You should perhaps also mention that $\sum_1^\infty a_n$ denotes the series $(s_n)$, whose limit when it converges also is denoted by $\sum_1^\infty a_n$. –  AD. May 24 '12 at 21:53
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@AD.: I think this is a bad convention for precisely the reason that one ought to distinguish a series from its sum. –  Qiaochu Yuan May 24 '12 at 22:14
    
That might be, but still it is very common. –  AD. May 25 '12 at 5:17

If you want to give your friend a visual approach, try this. Draw a square. Bisect it vertically and fill in the left side (that corresponds to $1/2$). Then bisect the right rectangle horizontally, and fill in the bottom ($1/2+1/4$). Bisect the unfilled square vertically, and fill in the left ($1/2+1/4+1/8$). Continue on like this to give your friend the general idea.

The reason that it is equal to $1$ (i.e.: to the whole square) is that for every point inside the square, we can iterate this procedure far enough so that that point gets shaded over. In other words, this procedure fills in the square, taken to the limit, so the corresponding area (number) is at least $1$. But at every stage, we are only filling in sections inside the square, so it is at most $1$, too, and thus, equal to $1$.

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I had a question closed a while back about something very similar - write the sum in base 2. It comes out as 0.1111111 ... Is this the same as 1?

Well, yes, because that's how limits are defined. But my daughter was struggling towards some language about open sets and limit points (therefore closed sets) and wanted 0.999999 (use base 10) to be different from 1 to indicate (effectively) that the set of partial sums did not include the limit point.

I reckon that a 13-year-old who can even think of conceptualising that kind of thing (no suggestion from me) is doing pretty well. Especially as this is one of the harder conceptual leaps between what most students get at High School and what they have to deal with at university.

The question is mathematically resolved, but the resolution is much more subtle than the indoctrinated elite (like me) sometimes imagine.

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