Nth-term test

The nth-term test states that an infinite series diverges if its terms fail to approach zero. Terms approaching zero is necessary for convergence but does not prove it.

Infinite series: An ordered sum of infinitely many terms, defined through the limit of its finite partial sums. The test diagnoses convergence of this sum by examining its individual terms.
Term test for divergence: A test declaring a series divergent when its terms fail to approach zero. This is the Nth-term test’s standard alternative name.
Ratio test: A series test based on the limit of the ratio of consecutive term magnitudes. Unlike the Nth-term test, it can prove convergence when terms approach zero.
Harmonic p-series: The family of series with terms one divided by n raised to a fixed exponent. For exponents at or below zero, its terms fail to vanish, so divergence follows immediately.
Sequence: An ordered list of numbers indexed by positive integers or another specified set. The series terms form a sequence whose limit the test checks.
Geometric series: A series whose successive terms have a constant ratio. Its terms tend to zero exactly when the ratio’s magnitude is below one, yielding convergence.
Root test: A series test based on the limit of the nth root of each term’s magnitude. It extracts exponential decay information that the term limit alone misses.
Telescoping series: A series whose terms cancel across partial sums, often leaving only boundary terms. The term test can reject examples before cancellation is analyzed when terms do not vanish.
Limit of a sequence: A value that sequence terms approach as their indices grow without bound. The test’s decisive condition is whether the term sequence has limit zero.
Harmonic series: The divergent series whose nth term is one divided by n. Its terms tend to zero, showing that the test cannot establish convergence.

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