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Sum of Infinite Geometric Series Calculator

Online infinite geometric series calculator: for |q|<1 the sum is S=a₁/(1−q), also used to convert repeating decimals to fractions.

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S=a_1/(1−q), |q|<1
First
Must have |q|<1

📖 Tutorial | Infinite Series

1. Definition
For |q|<1 the partial sums approach S=a₁/(1−q); for |q|≥1 the series diverges.
2. Formula & Notation
SInfinite sum (limit)
3. Properties & Laws
  • Converges for |q|<1;
  • Diverges for |q|≥1;
  • 0.3̄=1/3;
  • 1+½+¼+…=2.
4. Steps
  1. Enter a₁, q;
  2. Click "Sum";
  3. Or click "Load Example".
5. Worked Examples
Example: q=½ gives S=1/(1−½)=2.
6. Common Pitfalls
Need |q|<1;
It is a limit of partial sums;
q=−1 oscillates.

❓ FAQ | Infinite Series

Condition?
|q|<1.
Formula?
a₁/(1−q).
|q|≥1?
Diverges.
0.3̄?
1/3.
0.9̄?
1.
Meaning?
Limit of partial sums.
Geometry?
Repeated subdivision.