Problem: (Stanford Math 51H, Cauchy Root Test) Suppose there exists a and an integer such that for all . Prove that converges.
Solution: Well let’s just discard because it is finite and obviously converges. It remains to show that converges.
But then we have . So
The last summation, however, is a geometric series with common ratio , so it converges. Hence our sum does as well. QED.
Comment: The root test is, in effect, a comparison to a geometric series. The hypothesis is that we can bound by a geometric series for all large , implying convergence.
Practice Problem: (Stanford Math 51H) Discuss the convergence of
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