Analyze series convergence using the ratio test, root test, and divergence test, or compute geometric series sums.
Last updated
Series Analysis
Analyze a series of terms or a geometric series for convergence using the ratio test, root test, and divergence test.
Verdict
Test Results
Divergence Test Inconclusive
Terms appear to approach 0 (tail average magnitude: 0.3875). This is necessary but not sufficient for convergence.
Ratio Test Converges
The ratio |a(n+1)/a(n)| approaches 0.5, which is less than 1.
Root Test Converges
The nth root |a(n)|^(1/n) approaches 0.626505, which is less than 1.
Calculus
A series convergence calculator analyzes infinite series using the divergence test, ratio test, and root test. It also computes geometric series sums when the common ratio is between −1 and 1.
The divergence test checks if terms approach zero — if they do not, the series must diverge. The ratio test computes lim |a_{n+1}/a_n|: if less than 1 the series converges absolutely, if greater than 1 it diverges.
For geometric series with first term a and common ratio r, the infinite sum equals a/(1−r) when |r| < 1.
S = a / (1 − r), |r| < 1
Geometric series sum formula. This is the specific relationship the calculator applies when building the result.
A worked example helps translate the series convergence calculator maths into a realistic scenario so the user can compare the headline result with a concrete set of inputs.
That matters because a result is easier to trust when the page shows how the same logic behaves in a practical case instead of leaving the formula abstract.
Use the series convergence calculator output as a planning aid, then compare it with the assumptions, units, and caveats shown elsewhere on the page before acting on the number alone.
That extra interpretation step matters because a calculator can simplify the arithmetic but still cannot replace real-world context such as local rules, contract terms, or individual circumstances.
Frequently asked questions
When the ratio test limit equals 1 the test is inconclusive and another method must be used.
Yes — the harmonic series 1/n has terms approaching zero but diverges. The divergence test only catches series whose terms do not approach zero.
The safest manual check is to follow the same formula or rule one step at a time and compare that working with the calculator output. That catches sign errors, bracket mistakes, and input-order mixups without requiring any extra method beyond the underlying maths itself.
Also in Calculus
Related
These related calculators come from the same leaf category, nearby sibling categories, or the same top-level topic.
Differentiate a polynomial-style expression in x with a derivative calculator that shows steps, first and second derivatives, slope at a point, concavity.
Use an integral calculator with steps for polynomial-style expressions, definite bounds, average value, midpoint and trapezoid checks.
Evaluate polynomial ratio limits using direct substitution and L'Hôpital's rule for indeterminate forms.
Calculate descriptive statistics from one dataset.
