Mathfolis

Geometric Series

Unit 10 · Infinite Sequences & Series

Geometric Series

A geometric series ∑arⁿ converges to a/(1−r) when |r| < 1, and diverges when |r| ≥ 1. The first term a is the value at the starting index — when the series begins at n = k instead of n = 0, set a = arᵏ.

Geometric Series

∑n=0∞arn=a1−r,∣r∣<1\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}, \quad |r|<1
Diverges if
∣r∣≥1|r| \ge 1
AP Tip: When the series starts at n = k, the first term is arᵏ (not a). Identify this carefully before applying the sum formula.
Caution: The sum formula gives the total from the first term — it does NOT give a partial sum.
Type 1

Convergence and Sum (n starts at 0)

Identify a and r, verify |r| < 1, then apply the sum formula S = a/(1−r).

Example 1

Find the sum of ∑[n=0 to ∞] 5·(2/3)ⁿ.

Example 2

Does ∑[n=0 to ∞] 2·(−1.1)ⁿ converge or diverge?

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Type 2

Non-Zero Starting Index

When the sum starts at n = k, the first term is arᵏ. Use that as a in the formula.

Example 3

Find the sum of ∑[n=3 to ∞] (1/4)ⁿ.

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Type 3

Repeating Decimals as Geometric Series

Write a repeating decimal as an infinite sum, identify a and r, then apply the formula.

Example 4

Express 0.272727… as a fraction using a geometric series.

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