How do you test the series #(lnk)/(k^2)# for convergence?
1 Answer
The series:
is convergent.
Explanation:
To test the convergence of the series:
we can use the integral test, with test function:
Verify that the hypotheses of the integral test theorem are satisfied:
(i)
#f(x)# is positive in#[1,+oo)# (ii)
#(df)/dx = (1-2lnx)/x^3# is negative in#[1,+oo)# so the function is monotone decreasing in the interval
(iii)
#lim_(x->oo) f(x) = 0# (iv)
#f(k) = lnk/k^2#
So the convergence of the series is equivalent to the convergence of the integral:
We start solving the indefinite integral by parts:
so:
The integral converges, so the series is also proven to be convergent.
