`sum_(n=2)^oo lnn` Determine the convergence or divergence of the series.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

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`

...

See
This Answer Now

Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

Sign Up for 48 Hours Free Access

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. 

Approved by eNotes Editorial Team