Geometric Series Convergence Checker

How it is calculated

The series is a + ar + ar² + ar³ + …, with first term a and common ratio r. The tool first reports |r| to six decimals. The series converges when |r| < 1, and then the sum is S = a / (1 − r). When it does not converge, the sum column shows what happens: for r ≥ 1 it shows ∞ if a is positive and −∞ if a is negative, and for r ≤ −1 it shows "diverges", because the partial sums swing back and forth (r = −1) or grow in alternating signs. A first term of 0 always counts as convergent with sum 0, whatever r is. Enter r as a plain decimal such as 0.5 or −0.5; fractions like 1/2 and percent signs are not accepted. Results are rounded to six decimals.

Common mistakes

• Typing the ratio as a fraction (1/2) or a percentage (50%). The field takes a decimal such as 0.5; text like these returns no result.
• Using the ratio between the first two terms of a series that is not geometric. The formula only applies when every term is the previous one multiplied by the same r, so check the third term too.
• Mixing up r with 1 − r. The sum is a divided by (1 − r), so for r = 0.9 the sum is ten times the first term, not close to a.
• Entering a negative ratio as its absolute value. With r = −0.5 the sum is 5.333333, while r = 0.5 gives 16.000000; the sign matters for the sum, though not for convergence.
• Reading the ∞ or "diverges" output as an error. For |r| = 1 or more the series really has no finite sum, and r = −1 never settles on one value.