Sequences and Series Convergence

Adding up infinitely many numbers sounds like it should always give infinity — and sometimes it does. But a perpetuity that pays a shrinking coupon forever, or a discounted cash flow stretching to an infinite horizon, adds up infinitely many terms and still lands on a perfectly finite price. Distinguishing "infinite sum that blows up" from "infinite sum that settles on a finite number" — and knowing exactly which is which for a given formula — is what the theory of sequences and series convergence provides.

An analogy: walking half the remaining distance, forever

Walk halfway to a wall, then halfway again, then halfway again, forever. You take infinitely many steps, but you never walk more than the original distance to the wall — the total distance covered adds up to a finite number (twice the first step), even though the number of steps is unbounded. That's the essence of a convergent series: infinitely many nonzero pieces, but a finite total, because each new piece shrinks fast enough that the running total stops growing meaningfully. A series that keeps adding roughly the same-sized chunks forever, by contrast, has no hope of a finite total.

The math, one symbol at a time

A sequence $a_1, a_2, a_3, \ldots$ converges to a limit $L$ if $\lim_{n\to\infty} a_n = L$ — the terms get arbitrarily close to $L$ as $n$ grows, in the same sense as an ordinary limit. A series is the sum of a sequence's terms, $\sum_{n=1}^{\infty} a_n$, and it converges to a value $S$ if its sequence of partial sums $S_N = \sum_{n=1}^{N} a_n$ converges to $S$ as $N \to \infty$. A key workhorse is the geometric series:

$$\sum_{n=0}^{\infty} r^n = \frac{1}{1-r} \quad \text{whenever } |r| < 1.$$

In words: adding up a quantity that shrinks by a constant factor $r$ each term forever, provided that factor is smaller than 1 in size, gives a finite total — and that total has this simple closed form. If $|r| \geq 1$, the terms don't shrink fast enough (or grow), and the series diverges to infinity.

Worked example 1: pricing a growing perpetuity

A perpetuity pays $100 next year, growing at 2% per year forever, discounted at 7%. Each year's discounted payment is $100 \times \frac{(1.02)^{n-1}}{(1.07)^n}$, which is a geometric series with ratio $r = \frac{1.02}{1.07} \approx 0.9533$. Since $|r|<1$, the series converges, and the price is \frac{100/1.07}{1 - 0.9533} = \frac{93.46}{0.0467} \approx \2001$. Every one of the infinitely many future cash flows is included, yet the total is a clean finite number — exactly because the discounted, growth-adjusted ratio$r$ stays below 1.

Worked example 2: checking convergence with the ratio test

Does $\sum_{n=1}^{\infty} \frac{n}{2^n}$ converge? Apply the ratio test: compute $\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n\to\infty} \frac{(n+1)/2^{n+1}}{n/2^n} = \lim_{n\to\infty} \frac{n+1}{2n} = \frac{1}{2}$. Since this limit is less than 1, the series converges (its actual sum happens to be exactly 2, though the ratio test only confirms convergence, not the value). Compare with $\sum \frac{1}{n}$ (the harmonic series): its ratio test limit is exactly 1 — inconclusive — and in fact this series diverges despite each term shrinking to zero, a classic case where "the terms go to zero" is necessary but not sufficient for convergence.

Watch a running partial sum settle toward its limiting value as more terms are added — for a convergent series the curve flattens out; for a divergent one it keeps climbing with no ceiling in sight.

What this means in practice

Every perpetuity or terminal-value formula in finance, the convergence of numerical methods used to price derivatives, and the theoretical justification for estimators improving as more data arrives (see The Law of Large Numbers) all rest on series and sequence convergence. Recognizing a formula as a geometric series is what makes perpetuity and growing-annuity valuation formulas derivable in closed form rather than needing an infinite sum computed numerically. It also matters for judging how fast a numerical method is worth trusting: a Monte Carlo price estimate or an iterative optimizer's running solution is itself a sequence, and knowing whether — and how quickly — that sequence converges tells you whether stopping after a finite number of iterations gives an answer close to the true limit, or whether you've stopped too early while the sequence is still wandering. The renewal reward theorem (see The Renewal Reward Theorem) leans on a related convergence idea, applying a law-of-large-numbers argument to long-run averages of repeating cycles rather than to a simple infinite sum.