Comparison test

The comparison test determines convergence or divergence of a series with nonnegative terms by bounding its terms against those of another series with known behavior.

Direct comparison test: A series test that compares nonnegative terms using an inequality between corresponding terms. This is the basic form: a smaller convergent series or larger divergent series settles the result.
Geometric series: A series whose successive terms have a constant ratio; it converges when the ratio’s absolute value is below one. Its known convergence threshold makes it a common comparison target.
Integral test: A series test that relates a positive decreasing sequence to the improper integral of a corresponding function. It offers another route when terms fit a decreasing function more naturally than a known series.
nth-term test for divergence: A test that proves divergence when a series’ terms fail to approach zero. Unlike comparison, it gives no convergence conclusion when terms do approach zero.
Limit comparison test: A series test that compares positive terms through the limit of their ratio. It extends comparison to terms whose sizes are asymptotically proportional rather than ordered term by term.
Harmonic series: The divergent series whose nth term is the reciprocal of n. Its divergence provides a standard lower-bound benchmark.
Ratio test: A series test that examines the limit of the ratio of consecutive absolute terms. It can settle cases where direct comparison is difficult, especially with factorial or exponential terms.
Conditional convergence: Convergence of a series that fails to converge absolutely. A positive-term comparison cannot capture convergence caused by cancellation of signs.
Convergence of a series: A series converges when its sequence of partial sums approaches a finite limit. The test establishes this property indirectly, without summing the target series.
p-series: A series of terms 1/n^p, convergent for p greater than one and divergent for p at most one. Its sharp threshold helps classify many series by comparison.

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