Parts b and f Apply d\'Alembert\'s ratio test to each series and state explicitl

Question

Parts b and f

Apply d'Alembert's ratio test to each series and state explicitly the conclusions which you draw from use of the test. In parts (a)-(e) make a separate investigation and decide whether the series converges or diverges for each of the values of x for which the ratio test is indecisive. xn / n2. n + 1/n2(n + 2) xn. n! / xn. 3.5 (2n + 1)/5 . 10 5n xn n!/nn xn. n!/3 . 5 (2n + 1) xn. np / 2n-1 xn. (2n)n/(n + 1)n+1 xn.

Explanation / Answer

The series vn converges if vn+1/vn = r, where r >=1 for all sufficiently large n, is trivial. Logarithmic test is used to test functions like E(1/nlogn)which tend to zero more rapidly than n^(-1), but less rapidly than any power n^(-1-a).

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