Geometric Series

Principle

A geometric series has terms made by multiplying by the same fixed ratio each time. This pattern is useful whenever a model has repeated fractional contributions, such as successive reflections, damping factors, or repeated percentage losses.

The main advantage of a geometric series is that its partial sums can be found by a short algebraic cancellation.

Notation

• \(a\) is the first coefficient in the geometric series.
• \(x\) is the common ratio between consecutive terms.
• \(n\) is the summation index.
• \(N\) is the number of terms in the finite partial sum when the index starts at \(0\).
• \(S_N\) is the finite geometric partial sum.
• \(|x|\) is the absolute value of the ratio \(x\).

Method

To derive the finite geometric sum, write the sum, multiply it by the ratio, then subtract.

Multiplying by \(x\) shifts every term one place:

Subtract the shifted line from the original line before dividing:

If \(|x|\lt1\), then \(x^N\to0\), so the infinite sum has a finite value.

Rules

• For \(x\ne1\), the finite sum is \(S_N=a(1-x^N)/(1-x)\).
• If \(x=1\), the finite sum is simply \(N\) copies of \(a\), so \(S_N=Na\).
• The infinite formula \(\sum_{n=0}^{\infty}ax^n=a/(1-x)\) is valid only when \(|x|\lt1\).
• If the terms do not approach \(0\), the infinite series diverges.

Examples

Compute \(\sum_{n=0}^{\infty}3(1/2)^n\). Here \(a=3\) and \(x=1/2\). Since \(|x|=1/2\lt1\), use the infinite geometric formula:

The finite derivation explains why this works. First write \(S_N=3+3/2+3/4+\cdots+3(1/2)^{N-1}\). Then \((1/2)S_N=3/2+3/4+\cdots+3(1/2)^N\). Subtracting leaves only the first and last shifted terms:

So \(S_N=6(1-(1/2)^N)\), and the term \((1/2)^N\) tends to \(0\).

Checks

• Check the first index: starting at \(0\) and starting at \(1\) give different first terms.
• Check \(|x|\lt1\) before using the infinite formula.
• Treat \(x=1\) separately, because the denominator \(1-x\) is zero.
• Do not use the infinite formula when \(|x|\ge1\).