A geometric sequence is built by multiplying by the same number r again and again.
If |r| is less than 1, each multiplication shrinks the terms toward 0; if it is greater than 1, every multiplication makes them bigger and they never settle on a value.
So whether lim rⁿ exists is decided by the ratio r alone, and the dividing lines are r = 1 and r = −1.
Move the ratio slider and the points for n = 1 to 12 gather at 0, leave the picture, or alternate between 1 and −1.
The Ratio r Decides the Limit of rⁿ
seegongsik.com
Values of rⁿ for n = 1 to 12; terms that leave the picture are shown as red points at the edge
0.7
👀 Four behaviors in the picture
①If |r| < 1, the size shrinks with every multiplication and the terms converge to 0. When r is negative, they approach 0 while flipping above and below the axis
②If r = 1, every term is 1
③If r = −1, the terms alternate between −1 and 1, so there is no limit (they oscillate)
④If |r| > 1, the size grows without bound and the sequence diverges. At r = 1.5, r¹² is already above 129
⑤The r¹² value in the top band also shows that a ratio close to 1, such as r = 0.9, approaches 0 only slowly
Four Cases
Limit of rⁿ
lim rⁿ by ratio r
Ratio r
lim rⁿ
Behavior
|r| < 1
0
Converges
r = 1
1
Converges
r = −1
None
Diverges (oscillates)
|r| > 1
None
Diverges
🧭 The dividing lines r = 1 and r = −1
①The boundary is |r| = 1
②The sequence converges exactly when −1 < r ≤ 1. The right end r = 1 is included and the left end r = −1 is not
③At r = −1 the terms keep size 1 but never settle on one value, so they do not converge
④Problems on limits of geometric sequences come down to where the ratio sits in this range
Condition for Convergence
When {rⁿ} converges
limn→∞ rn converges ⇔ −1 < r ≤ 1
Only in this range does the limit exist: it is 0 when −1 < r < 1 and 1 when r = 1
The limit used most
−1 < r < 1 ⇒ limn→∞ rn = 0
Raising a number smaller than 1 in absolute value to higher and higher powers brings it closer and closer to 0
Limits of Fractions: Divide by the Fastest-Growing Term
✏️ Limits of Fractions: Divide by the Fastest-Growing Term
When the numerator and denominator mix terms of the form aⁿ, divide both by the term whose base is largest in absolute value. The other terms then become (smaller base/larger base)ⁿ and go to 0.
Example 1
For r>1, find limn→∞rn - 1rn + 1.
1
Divide the numerator and denominator by the fastest-growing term, rⁿ.
rn - 1rn + 1 = 1 - (1/r)n1 + (1/r)n
2
If r > 1, then 0 < 1/r < 1, so (1/r)ⁿ → 0.
→ 1 - 01 + 0 = 1
▸ 1
Finding the fastest-growing term and dividing by it is the standard method for limits of fractions.