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Whether Int f(x)dx, x = 2 to x = infinite, converges, then the series Sum 1/n*ln n converges.
We'll find out 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 determine 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 series Sum 1/n*ln n, if n=2 to n = infinite, is divergent.
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