How to use the integral test to determine whether ∫ d x ln x converges or diverges from [ 2 , ∞ ) ?

Question

How to use the integral test to determine whether       ∫                  d        x                    ln        x             converges or diverges from             [      2      ,      ∞      )      ?

Alexis West · Accepted Answer

Note that in the interval       x    ∈          [      2      ,      ∞      )       the function:            f              (        x        )              =          1              ln        x             is:1) Infinitesimal as             lim              x        →        ∞                    f              (        x        )              =    0  2) Positive as             f              (        x        )              &amp;gt;    0   for       x    &amp;gt;    1  3) Decreasing. In fact       f    ′          (      x      )        =    -          1              x                              ln            2                    x                      &amp;lt;    0  4)             f              (        n        )              =          1              ln        n            Therefore, according to the integral test, the integral&#039;s convergence:                    ∫        2        ∞                            d        x                    ln        x             is equivalent to the convergence of the series:                    ∑                  n          =          2                ∞                    1              ln        n            Now we can easily demonstrate that:            ln      n        &amp;lt;    n  so that:            1              ln        n              &amp;gt;          1      n      and as we know that the harmonic series:                    ∑                  n          =          1                ∞                    1      n      is divergent, we can conclude that:                    ∑                  n          =          2                ∞                    1              ln        n            is divergent by direct comparison, and hence also:                    ∫        2        ∞                            d        x                    ln        x             is divergent.

How to use the integral test to determine whether int dx/lnx converges or diverges from [2,oo)?

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