Algebra 2
Sequences and series
Infinite geometric series
20 practice questions
0 video lessons
Theory + worked examples
Theory
An infinite geometric series adds infinitely many terms. It converges only when the ratio is small:
\[S=\dfrac{a_1}{1-r}\quad\text{if}\quad |r|<1.\]
If \(|r|\ge1\), the terms don't shrink and the series diverges.
Only shrinking terms (\(|r|<1\)) add to a finite total.
Partial sums approach the limit \(16\).
When an infinite series converges.
Sum to infinity:
\[S=\dfrac{a_1}{1-r},\qquad |r|<1\]
Check \(|r|<1\) first — otherwise there is no sum.
How to sum an infinite series
- Find \(a_1\) and \(r\).
- Check that \(|r|<1\).
- Apply \(S=\dfrac{a_1}{1-r}\).
- If \(|r|\ge1\), state that it diverges.
Example 1 — A convergent sum
Find \(8+4+2+1+\dots\).
Solution
Here \(a_1=8,\ r=\dfrac12\), and \(|r|<1\).
| \(S\) | \(=\) | \(\dfrac{8}{1-\dfrac12}\) |
| \(=\) | \(\dfrac{8}{\dfrac12}=16\) |
Example 2 — Repeating decimal
Express \(0.\overline{9}\) as a fraction using a series.
Solution
\(0.9+0.09+\dots\) has \(a_1=0.9,\ r=0.1\).
| \(S\) | \(=\) | \(\dfrac{0.9}{1-0.1}\) |
| \(=\) | \(\dfrac{0.9}{0.9}=1\) |
Example 3 — Does it converge?
Does \(2+4+8+16+\dots\) have a finite sum?
Solution
The ratio \(r=2\) has \(|r|\ge1\).
| \(|r|\) | \(=\) | \(2\ge1\) |
It diverges — no finite sum.
Example 4 — Find the ratio limit
Find \(1+\dfrac13+\dfrac19+\dots\).
Solution
\(a_1=1,\ r=\dfrac13\).
| \(S\) | \(=\) | \(\dfrac{1}{1-\dfrac13}\) |
| \(=\) | \(\dfrac{1}{\dfrac23}=\dfrac32\) |
Common pitfalls
Converges only if \(|r|<1\) — always check first.
Use \(a_1\), the first term, in the formula.
\(|r|\ge1\) means no finite sum.
Frequently asked questions
When does an infinite geometric series converge?
When \(|r|<1\).
What is the sum to infinity?
\(S=\dfrac{a_1}{1-r}\).
What happens if \(|r|\ge1\)?
The series diverges — there is no finite sum.
Why does \(0.\overline9=1\)?
The geometric series \(0.9+0.09+\dots\) sums to exactly \(1\).
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