For what r does 3/(n^(2r - 3))-3/n converge or diverge?

1 Answer
May 29, 2016

The series sum converges only for r = 2

Explanation:

This series can be decomposed as the sum of two series: sum_n a_n^1 = sum_n 3/(n^{2r-3} and sum_n a_n^2 = sum_n -3/n so that sum_n a_n = sum_n (a_n^1+a_n^2). The series sum_n a_n^2 is allways divergent and sum_n a_n^1 converges or diverges depending on r. So sum_n a_n is convergent only for r = 2 when a_n^1+a_n^2=0