What is the interval of convergence of sum_1^oo (3n)/n^(x)?

1 Answer
Aug 4, 2016

x>2

Explanation:

For x-1>0 which is the condition for which k^{-(x-1)} decreases monotonically, we have sum_1^n 3 k^{-(x-1)} le int_1^n 3k^{-(x-1)}dk +3= n^{2-x}/(2-x)-1/(2-x)+3 and also for 2-x < 0 we have

lim_{n->oo}n^{2-x}/(2-x)-1/(2-x)+3 = 1/(x-2)+3. So
we have

sum_1^n 3 k^{-(x-1)} le 3/(x-2)+3