Geometric Series Sum Calculator
Find the sum of a finite geometric series or an infinite geometric series, with convergence checks, nth-term output, first terms, formulas, and worked examples.
Geometric series calculator for finite and infinite sums
A geometric series is the sum of a geometric sequence. If the sequence is 4, 12, 36, 108, the series is 4 + 12 + 36 + 108. This calculator handles both finite sums, where you add a fixed number of terms, and infinite sums, where the terms continue forever but may still approach a finite total.
OpenStax describes a geometric series as the sum of terms from a geometric sequence in its geometric sequences and series lesson. The key idea is that each term is created by multiplying the first term by a power of the common ratio.
Geometric series formulas
The finite formula works whether the ratio is bigger than 1, between 0 and 1, negative, or zero. The infinite formula is more restrictive. It only works when the absolute value of the ratio is less than 1, because only then do the terms shrink toward zero.
Finite vs infinite geometric series
A finite geometric series stops after n terms, so it always has a calculable sum. An infinite geometric series continues forever. It has a finite sum only if the terms become smaller and smaller fast enough. For example, 10 + 5 + 2.5 + 1.25 + ... converges because r = 0.5. But 10 + 20 + 40 + 80 + ... does not converge because r = 2.
| Ratio r | Finite series | Infinite series | Behavior |
|---|---|---|---|
| 0 < r < 1 | Works | Converges | Positive terms shrink |
| -1 < r < 0 | Works | Converges | Signs alternate and shrink |
| r ≥ 1 | Works | Diverges | Terms do not shrink |
| r ≤ -1 | Works | Diverges | Terms alternate but do not shrink |
Worked example: finite geometric series
Because this is a finite sum, it does not matter that r is greater than 1. You are only adding eight terms, so the total is a fixed number. The convergence restriction applies only to infinite series.
Worked example: infinite geometric series
The terms continue forever, but each new term is only one-quarter of the previous term. The remaining tail becomes so small that the total approaches 16.
Common questions
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Use S_n = a_1(1 - r^n)/(1 - r) when r is not equal to 1. Here a_1 is the first term, r is the common ratio, and n is the number of terms. If r = 1, every term is the same, so the sum is simply n times a_1. A finite geometric series always has a sum because it stops after a specific number of terms.
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An infinite geometric series converges only when the absolute value of the common ratio is less than 1. In symbols, -1 < r < 1. That condition means the terms shrink toward zero. If the terms do not approach zero, the infinite total cannot settle to a finite number.
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S_n means the sum of the first n terms, so it is a finite sum. S infinity means the sum of all terms forever, so it is an infinite sum. S_n can be calculated for any common ratio. S infinity only exists for a geometric series when the common ratio is between -1 and 1.
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Yes. A negative ratio makes the terms alternate signs. For a finite series, the formula still works normally. For an infinite series, a negative ratio is allowed only if it is greater than -1 and less than 0. For example, r = -0.5 converges, but r = -2 does not.
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If r = 2, the terms double forever instead of shrinking. The total grows without bound, so there is no finite number that the series approaches. The formula S infinity = a_1/(1 - r) is only valid after the convergence condition is satisfied. Using it when |r| is greater than or equal to 1 gives a meaningless result for an infinite series.
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If r = 1, every term is equal to the first term. A finite sum is n times the first term. An infinite series with r = 1 does not converge unless the first term is zero, because the terms do not shrink toward zero. This is why the usual finite formula has a separate case for r = 1.
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Divide any term by the term before it. For example, in 6, 18, 54, 162, the common ratio is 18 ÷ 6 = 3. You should check at least one more pair if several terms are given. If the ratios are not the same, the series is not geometric and the geometric series formula should not be used.
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Yes. Many repeating decimals can be written as infinite geometric series because each repeated decimal place is a fixed fraction of the previous place value. For example, 0.333... can be viewed as 0.3 + 0.03 + 0.003 + ..., where the common ratio is 0.1. The infinite geometric sum explains why that total equals 1/3.