What is the sum of the **infinite** geometric series $1 + \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \dfrac{1}{16} + \ldots$?

First confirm geometric, with $a_{1} = 1$ and $r = \dfrac{1}{2}$. Since $|r| = 1/2 < 1$, the infinite sum converges: $S_{\infty} = \dfrac{a_{1}}{1 - r} = \dfrac{1}{1 - 1/2} = \dfrac{1}{1/2} = 2$. Intuition: each successive term halves the remaining gap to $2$. After $1 + 0.5 = 1.5$, you're $0.5$ short. After $+0.25$, you're $0.25$ short. The gap halves forever but never goes below zero. - Trap A ($1$) returns just the first term. - Trap B ($1.5$) returns the partial sum after two terms. - Trap D would apply if $|r| \ge 1$. This series ($1 + 1/2 + 1/4 + \ldots = 2$) is the most famous example of a convergent infinite geometric series and underlies Zeno's paradox.