Infinite Series & Sums

Medium3–4 hrsGeometric SeriesTelescopingRecursionConvergence

Overview

Infinite series appear constantly in probability (geometric distributions, expected values of random sums) and in finance (discounted cash flows, perpetuities). The geometric series is the foundation. The self-referential trick — 'S equals something that involves S' — is particularly elegant and appears in many expected-value recursions. Mastering these series gives you quick answers to problems others grind through.

How to Recognize

→Expected number of trials that follow a geometric pattern
→'What does this series converge to?'
→Recursion with a constant ratio < 1
→Self-referential: 'S equals this expression that involves S'

Step-by-Step Approach

1.Geometric: 1 + r + r² + … = 1/(1−r) for |r| < 1
2.Differentiate to get Σ n·r^(n−1) = 1/(1−r)²
3.Self-referential: set up S = a + r·S → S = a/(1−r)
4.Telescoping: write aₙ = f(n)−f(n+1), sum collapses to f(1)−f(∞)

Key Formulas

\sum_{n=0}^\infty r^n = \frac{1}{1-r}

for

|r|<1

\sum_{n=0}^\infty n\,r^n = \frac{r}{(1-r)^2}

\sum_{n=0}^\infty n^2 r^n = \frac{r(1+r)}{(1-r)^3}

04Geometric sum (finite):

\sum_{n=0}^{N} r^n = \frac{1-r^{N+1}}{1-r}

Worked Examples

0Quick Example

Win $1, with prob 1/2 get another go (recursively). S = 1 + (1/2)S → S = 2.

Problem

Derive E[geometric] using the series formula. P(heads) = p.

Solution

E[flips] = 1·P(H) + 2·P(TH) + 3·P(TTH) + … = Σₙ₌₁^∞ n·(1−p)^(n−1)·p.

Factor out p: E = p · Σₙ₌₁^∞ n·q^(n−1) where q = 1−p.

Use the derivative formula: Σ n·q^(n−1) = 1/(1−q)² = 1/p².

E = p · 1/p² = 1/p. ✓

Answer

$E[\text{geometric}] = 1/p$ . The series formula gives it directly: $\sum_{n=1}^\infty n q^{n-1} = 1/(1-q)^2 = 1/p^2$ .

Common Mistakes

!
Forgetting the convergence condition: geometric series only converges if |r| < 1.
!
Off by one: is the first term r^0 = 1 or r^1 = r? Sum from n=0 vs n=1.
!
For derivatives: Σ n·r^n = r·Σ n·r^(n-1) = r/(1-r)². The extra factor of r matters.

Practice Problems

Click "Show Answer" to reveal

Sum: 1/2 + 2/4 + 3/8 + 4/16 + … = Σ n/2^n

Hint: Use Σ n·r^n = r/(1-r)² with r=1/2.

Expected total profit if each trade independently makes $1 profit with probability 2/3 or$ 0 with probability 1/3, and you stop after your first $0 trade.

Hint: E[trades before stopping] = geometric mean. Each makes $1.

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