Analyze series convergence using the ratio test, root test, and divergence test, or compute geometric series sums.
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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.
Also in Calculus
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.
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.
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