See the question.
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By the limit comparison test if lim n-> infinity of a(n) / b(n) converges to nonzero real number then the series a(n) converges iff series b(n) converges
Here a(n) is (n+1)/n^(5/2)
Let b(n) = 1 / n^(3/2), then a(n)/ b(n) = [(n+1) / n^(5/2)][1 / n^(3/2)] = (n+1) / n
--> lim n-> infinity a(n) / b(n) = lim n-> infinity (n+1) n = 1
The ratio converges to a nonzero number and by the p-series test our series b(n) converges, hence by the limit comparison test our series converges
Here a(n) is (n+1)/n^(5/2)
Let b(n) = 1 / n^(3/2), then a(n)/ b(n) = [(n+1) / n^(5/2)][1 / n^(3/2)] = (n+1) / n
--> lim n-> infinity a(n) / b(n) = lim n-> infinity (n+1) n = 1
The ratio converges to a nonzero number and by the p-series test our series b(n) converges, hence by the limit comparison test our series converges