The ratio test detects eventual geometric shrinking or growth.
WHAT HAPPENS IF…What does a ratio limit of 1 tell us by itself?
Interactive exploration
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Connect the picture to the ideaan=nkrn,anan+1=r(1+n1)k→r
For positive r and nonnegative integer k, a ratio limit below 1 proves convergence; above 1 proves divergence. A limit of 1 makes the ratio test inconclusive. In this specific family r=1 gives terms nᵏ, which do not tend to zero, so another test proves divergence.
A detail that matters
One finite ratio above 1 does not determine the limiting ratio.