Geometric Series Calculator
Geometric Series Calculator
Slug: `/geometric-series-calculator/`
Primary keyword: geometric series calculator
Meta title: Geometric Series Calculator: Exact Sum When the Ratio Is Under One
Meta description: Find whether a geometric series converges and get its exact sum. Handles any starting index, with the common ratio and closed form shown.
Word count: approximately 1,000
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The one series that always gives you a number
Most convergent series converge to constants nobody can write down. Geometric series are the exception, and that makes them the backbone of the whole subject.
A series is geometric when each term is a fixed multiple of the one before. Call that multiple r, the common ratio.
Convergence happens exactly when the absolute value of r is under 1. And when it converges, the sum has a formula.
The formula, and the detail people get wrong
Summing from n equal to 0, the total is a divided by 1 minus r, where a is the first term.
Summing from n equal to 1 instead, the total is a r divided by 1 minus r.
Those differ, and mixing them up is the most frequent error here. Always check where the index starts.
Worked through: the sum of one over two to the n, starting at n equal to 1. First term is 0.5, ratio is 0.5, so the sum equals 0.5 divided by 0.5, which is exactly 1. Starting the same series at n equal to 0 gives 2 instead.
Why the ratio decides everything
Write out a partial sum of the first N terms and you get a closed form: a times one minus r to the N, all over one minus r.
Now let N grow. When the absolute value of r is under 1, r to the N collapses to zero and the partial sum settles at a over one minus r. When the absolute value of r is at least 1, r to the N does not vanish, and nothing settles.
That is the entire proof, and it is unusual in this subject for being both short and complete.
Boundary cases
At r equal to 1 every term equals a, and the partial sums grow without limit unless a is zero.
At r equal to minus 1 the partial sums oscillate between a and 0 forever. No limit exists, so it diverges, even though the terms never grow.
Both cases fail the divergence test, since the terms do not approach zero.
Test in one line
Divide any term by the previous one. Constant answer means geometric.
Where they turn up
Zeno's paradox. Crossing a room by repeatedly halving the remaining distance is a geometric series with r equal to one half. It sums to a finite total, which is why the far wall is reachable.
Repeating decimals. The decimal 0.333 recurring is three tenths plus three hundredths plus three thousandths, a geometric series with r equal to 0.1. Its sum is one third, exactly.
Perpetuities. A payment that continues forever has finite present value because each future payment is discounted by a factor under 1. Same mathematics, different vocabulary.
Taylor series. The expansion of one over one minus x is the geometric series itself, and it holds only for absolute x under 1, which is where the radius of convergence idea starts.
Recognising one in disguise
Divide any term by the one before it. A constant result means geometric.
- Four to the n over five to the n is geometric with r equal to 0.8, so it converges to 4 from n equal to 1.
- Two to the n over three to the n plus one has ratio two thirds, and converges.
- n over two to the n is not geometric, since the ratio depends on n. It still converges, by the ratio test.
That last example matters. Many convergent series look geometric without being geometric, and only geometric ones hand you an exact sum.
Limits of this tool
Our calculator identifies a constant ratio and returns the exact sum. When the ratio drifts with n, it says so and switches to a general convergence test instead, which produces a verdict without a sum. Series that are geometric only after algebraic rearrangement may not be recognised. Simplify before entering them.
Questions
What if the absolute value of r equals exactly 1?
It diverges, either growing without bound or oscillating forever.
Can r be negative?
Yes. Convergence needs the absolute value under 1, so r equal to minus 0.5 converges fine, alternating as it goes.
Does the starting index change convergence?
No, only the sum. Convergence depends on r alone.
How do I convert a repeating decimal into a fraction?
Treat the repeating block as the first term and use the formula. For 0.333 recurring: first term 0.3, ratio 0.1, sum equals one third.
Why is the geometric series so central?
The ratio and root tests both work by comparing your series to a geometric one, so it is the yardstick everything else is measured against.