Geometric and Telescoping Series

A geometric series multiplies by a fixed ratio r each term. It converges exactly when |r| < 1.

The sum

When it converges, the sum is a ÷ (1 − r), where a is the first term.

Where that comes from

In the finite sum a(1 − rⁿ) ÷ (1 − r), the rⁿ shrinks to zero when |r| < 1, leaving a ÷ (1 − r).

Which term is a

a is the first term of the series as written, not necessarily the term at n = 0. Misreading it is the usual error.

Telescoping series

A telescoping series has terms that cancel in pairs. Writing out the first few partial sums shows what survives.

Find the partial sum

For a telescoping series, find a closed form for the partial sum, then take its limit. That is the whole method.

Geometric series converge when |r| < 1

The sum is a/(1 − r), valid only when |r| < 1. Outside that range the terms do not shrink and the series diverges. Checking the ratio before applying the formula is mandatory.

Where the formula comes from

Subtracting r times the partial sum from the partial sum collapses almost every term, leaving a short expression. Taking the limit gives a/(1 − r). The trick recurs throughout series work.

Telescoping series collapse

When each term splits into a difference, consecutive parts cancel and only the ends survive. Partial fractions is usually what produces the split, so the two techniques are used together.

Write out the partial sum

Expand several terms explicitly and see exactly what cancels and what remains before taking the limit. Assuming everything cancels except the first term is a common and wrong shortcut.