Use the integral test to determine the convergence of the series Sum 1/n*ln n where n=2 to n = infinite .

giorgiana1976 | Student

We know that if:

Int f(x)dx, x = 2 to x = infinite,

converges, then the series Sum 1/n*ln n converges.

We'll determine the integral:

Int f(x)dx = Int dx/x*ln x, x = 2 -> x = infinite

lim Int dx/x*ln x, for x = 2 to x = N, N->infinite

We'll evaluate first the indefinite integral:

Int dx/x*ln x

We'll put ln x = t. Differentiating both sides, we'll get:

dx/x = dt

Int dx/x*ln x  = Int dt/t

Int dt/t = ln t + C

Lim Int dx/x*ln x = lim ln (ln x)

lim ln (ln x) = lim [ln(ln N) - ln(ln 2)], N-> infinite

lim ln (ln x) = infinite

The given series Sum 1/n*ln n, where n=2 to n = infinite, is divergent.