The root test measures the exponential size of a sequence of terms.
WHAT HAPPENS IF…Why can the test work well for terms raised to the nth power?
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Connect the picture to the ideaan=[r(1+n1)]n,nan→r
A root limit below 1 bounds the tail by a convergent geometric series; above 1 the terms fail to approach zero. At r=1, the root test is inconclusive, but this particular sequence tends to e, so its series diverges by the term test.