TITLE: Does the fact that $\sum_{n=1}^\infty 1/2^n$ converges to $1$ mean that it equals $1$?
QUESTION [10 upvotes]: 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?

REPLY [8 votes]: 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.