The divergence test states that a series 
diverges if the limit
of 
as 
is nonzero or does not exist. If the series converges,
consecutive partial sums approach the same limit,
so their difference 
tends to 0. The converse fails for the harmonic series,
whose terms approach 0 even though the series diverges.
Divergence Test
See also
Convergence Tests, Convergent Series, Dini's Test, SeriesExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Divergence Test." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DivergenceTest.html