Andromeda
Note

Geometric Series

Definition

A Geometric Series is the sum of the terms of a geometric sequence.

Why It Matters

Geometric series calculate the cumulative effect of compounding over time; whether finding the long-term return on an investment or assessing convergence in physics, understanding how a constant ratio sums up is the difference between predicting infinity or a finite limit.

Core Concepts

  • Finite Geometric Series: The sum of the first nn terms is: Sn=∑k=1na1rk−1=a1(1−rn1−r),r≠1S_n = \sum_{k=1}^{n} a_1 r^{k-1} = a_1 \left( \frac{1 - r^n}{1 - r} \right), \quad r \neq 1

    • How to read: “The sum S n is equal to the sum from k equals one to n of a one times r raised to the quantity k minus one, which equals a one times the quantity one minus r to the n, all divided by the quantity one minus r.”
    • Meaning / when to use: Closed-form sum of nn geometric terms — avoids adding them one by one.

  • Infinite Geometric Series: If the common ratio ∣r∣<1|r| < 1, the series converges to a finite sum: S=∑n=1∞a1rn−1=a11−rS = \sum_{n=1}^{\infty} a_1 r^{n-1} = \frac{a_1}{1 - r}

    • How to read: “The sum S is equal to the first term a one divided by the quantity one minus r.”
    • Meaning: When ∣r∣<1|r| < 1, terms shrink fast enough that the infinite sum is finite — classic result for perpetuities and Zeno-type problems.

  • Divergence: If ∣r∣≥1|r| \geq 1 and a1≠0a_1 \neq 0, the infinite series diverges (the sum goes to infinity or oscillates).

    • How to read: “The absolute value of r is greater than or equal to one.”
    • Meaning: Terms do not shrink to zero fast enough; the partial sums grow without bound or oscillate.

Connected Concepts

Connected notes