Series, Sums, and the Geometric Reflex

A surprising number of interview questions reduce to recognizing a sum you already know in closed form. This lesson is the short catalog, plus the approximation tricks that make you fast without a calculator.

The geometric series — finance's favorite sum

$$\sum_{k=0}^{\infty} ar^k = \frac{a}{1-r} \quad (|r|<1), \qquad \sum_{k=0}^{n-1} ar^k = a\,\frac{1-r^n}{1-r}$$

Everywhere you look, it's this sum wearing a costume:

• A perpetuity paying $C$ per year at discount rate $r$: value $= \frac{C/(1+r)}{1 - 1/(1+r)} = \frac{C}{r}$. "What's \$100/year forever worth at 5$2,000, instantly.
• Expected trials to first success (probability $p$): differentiating the geometric series gives $E = 1/p$ — the "6 rolls to a six" fact, from first principles.
• Repeating decimals, tick-decay models, momentum half-lives — anything that shrinks by a constant factor.

The derivation trick is worth knowing cold: let $S = \sum ar^k$, compute $S - rS$, everything telescopes. Interviewers accept the formula; they're delighted by the two-line derivation.

The Gauss sums

$$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}, \qquad \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$$

The first is pairing (1+100, 2+99, …). The second explains, among other things, why the variance of a discrete uniform die is $\frac{n^2-1}{12}$. Also useful: the sum of the first $n$ odd numbers is $n^2$ — the noodle-problem sum $\sum \frac{1}{2k-1}$ starts to feel familiar once odd numbers do.

Harmonic sums and logarithms

$$H_n = \sum_{k=1}^{n} \frac{1}{k} \approx \ln n + 0.577$$

$H_n$ appears whenever "the $k$-th step succeeds with probability $1/k$": coupon collector ($E = nH_n$), records in a random sequence (expected number of running maxima in $n$ draws is $H_n$), the 100-prisoners bound. Knowing $H_{100} \approx 5.19$ turns several hard-looking questions into arithmetic.

Approximation kit

• $e^x \approx 1 + x + \frac{x^2}{2}$ and $\ln(1+x) \approx x - \frac{x^2}{2}$ for small $x$: the engine behind return compounding.
• $(1+x)^n \approx e^{nx}$: "2% growth for 30 periods" $\approx e^{0.6} \approx 1.82$.
• Rule of 72: doubling time at rate $r\%$ is $\approx 72/r$ periods (from $\ln 2 \approx 0.693$, with the numerator nudged to 72 for divisibility). At 8%: nine years. Interviewers use this constantly and expect you to.
• Useful constants at your fingertips: $\ln 2 \approx 0.69$, $e \approx 2.718$, $1/e \approx 0.37$ (secretary problem!), $\sqrt{2} \approx 1.414$, $\sqrt{252} \approx 15.9$ (daily→annual vol).

The interview version

*"A stock pays \$5/year forever, rates are 4$5/0.04 = 125$. *"Now the payment grows 2$\frac{1.02}{1.04}$: value$= \frac{5}{0.04 - 0.02} = 250. *"Estimate how long \1 takes to become \$10 at 7$\ln 10 \approx 2.3$, so$2.3/0.07 \approx 33$ years (or: three doublings is 8×, a bit more than three rule-of-72 periods ≈ 31 — both accepted). Fast, structured, and every step namable: that's the standard being tested.