Does the fact that $\sum_{n=1}^\infty 1/2^n$ converges to $1$ mean that it equals $1$?

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?

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.

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 $1$.

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.