Limit of a Geometric Sequence

The Fate of rⁿ Is Set by r

Common ratio r0.7

👀 See It

①If |r|<1, each multiplication shrinks toward 0

②If r=1, it stays 1

③If r=-1, it oscillates between 1 and -1 (diverges)

④If |r|>1, it blows up and diverges

Four Cases

Classifying lim rⁿ

📊lim rⁿ by common ratio r

Ratio r	lim rⁿ	Behavior
\|r\|<1	0	Converges
r=1	1	Converges
r=−1	None	Oscillates·diverges
\|r\|>1	None	Diverges

🧭 The Borders Are r=1 and r=-1

①The split happens at |r|=1

②Convergence holds only on the single band −1<r≤1

③r=−1 oscillates, so it is not convergence

④This boundary sense is the key to every geometric-limit problem

The Convergence Condition in One Line

Convergence condition for rⁿ

lim_n→∞ rⁿ converges ⇔ -1 < r ≤ 1

The limit exists only on this range (0 if −1<r<1, 1 if r=1)

Key limit value

-1<r<1 ⇒ lim_n→∞ rⁿ = 0

A ratio with absolute value below 1 vanishes to 0 under repeated powers

Rational Limits — Divide by the Dominant Term

✏️ Rational Limits — Divide by the Dominant Term

When rⁿ appears in numerator and denominator, divide by the term with the largest |r| to create parts that go to 0.

Example 1

For r>1, find lim_n→∞ rⁿ - 1rⁿ + 1.

Divide numerator and denominator by the dominant term rⁿ.

rⁿ - 1rⁿ + 1 = 1 - (1/r)ⁿ1 + (1/r)ⁿ

Since r>1 gives 1/r<1, we have (1/r)ⁿ→0.

→ 1 - 01 + 0 = 1

▸ 1

Deciding "which term grows faster" and dividing by it is the standard move for rational limits.

Wrap-up

Geometric-sequence limit summary

lim_n→∞ rⁿ = 0 (−1<r<1), 1 (r=1), diverges (otherwise)

Convergence on −1<r≤1; within it only r=1 gives 1, the rest give 0

2022 CSAT Math (Calculus) #23, adapted

Find lim_n→∞ 3ⁿ⁺¹ + 2ⁿ3ⁿ - 2ⁿ.

①0

②1

③2

④3

⑤Diverges

▸ ④ 3

Divide numerator and denominator by the largest base 3ⁿ.

3·3ⁿ + 2ⁿ3ⁿ - 2ⁿ = 3 + (2/3)ⁿ1 - (2/3)ⁿ

Since (2/3)ⁿ→0, the limit is 3/1.

→ 3 + 01 - 0 = 3

🎯 Exam Points

①lim rⁿ converges only on −1<r≤1

②For rational forms, divide by the largest base

③Engineer (small base/large base)ⁿ→0

④Do not forget r=−1 oscillation is divergence

⑤If n sits in an exponent, suspect a geometric-sequence limit first

Was this helpful? Support seegongsik