Geometric series test
Determine convergence and sum of an infinite geometric series.
Geometric series test is one of 6 series formulas in the calculus section of this library, and it is used at ap · university level.
Why geometric series test works
Multiply the whole sum by the ratio and subtract it from the original. Everything cancels except the first term, leaving the sum times (1 − r) equal to that first term. Solving gives the closed form, and the cancellation only settles down when the terms are shrinking.
What each symbol means
$a$ is the first term and $r$ the common ratio.
Geometric series test: when it holds
The series diverges when $|r|\ge1$.
When it stops applying
Once the ratio reaches 1 in size the terms stop shrinking and there is no sum at all. The formula still hands back a number, and the doubling series 1 + 2 + 4 and so on appears to total −1, which is a clear warning that the condition was skipped.
Geometric series test: a worked example
$1+1/2+1/4+\cdots=1/(1-1/2)=2$.
The mistake to avoid
What people do: Using the closed form with the wrong first term when the series starts at index 1 instead of index 0.
Why it goes wrong: The first term you actually add is then the original one multiplied by the ratio, so the standard form overshoots the true sum.
Do this instead: Identify the very first term that gets added and put that value into the formula.
Geometric series test: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a$ is the first term and $r$ the common ratio.
- Check the conditions before substituting. The series diverges when $|r|\ge1$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Series
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where geometric series test comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Sequences and Series — the lesson behind this formula: recognize arithmetic, geometric, and recursive patterns.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about geometric series test
What happens when the ratio is exactly 1 or exactly −1?
At 1 the terms never shrink and the running total runs away; at −1 the running total flips between two values forever without ever settling.
How do I find the ratio from a written-out series?
Divide any term by the one before it. If that answer is not the same every time, the series is not geometric and the formula does not apply.
Can the ratio be negative?
Yes. The terms then alternate in sign, and as long as the size of the ratio stays under 1 the closed form works exactly as written.
Why does 0.999 repeating equal 1?
It is geometric with first term 0.9 and ratio 0.1, so the closed form gives 0.9 divided by 0.9, which is exactly 1 and not merely close to it.