To evaluate the series `sum_(n=2)^oo ln(n),` we may apply the divergence test:
If `lim_(n-gtoo) a_n != 0` then `sum a_n` diverges.
From the given series `sum_(n=2)^oo ln(n)` , we have `a_n=ln(n)` .
Applying the diveregence test,we determine the convergence and divergence of the series using the limit:
`lim_(n-gtoo)ln(n) = oo`
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To evaluate the series `sum_(n=2)^oo ln(n),` we may apply the divergence test:
If `lim_(n-gtoo) a_n != 0` then `sum a_n` diverges.
From the given series `sum_(n=2)^oo ln(n)` , we have `a_n=ln(n)` .
Applying the diveregence test,we determine the convergence and divergence of the series using the limit:
`lim_(n-gtoo)ln(n) = oo`
When the limit value (L) is `oo ` then it satisfies `lim_(n-gtoo) a_n != 0 .`
Therefore, the series `sum_(n=2)^oo ln(n)` diverges.
We can also verify this with the graph: `f(n) = ln(n)` .
As the value of `n` increases, the function value also increases and does not approach any finite value of L.