solute if n(rootn 11-rootn5) convergent? exact answer pls

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to check if the function `n(root(n) 11-root(n)5) ` converges.

Evaluating the limit of the function yields:

`lim_(n-gtoo) n(root(n) 11-root(n)5) = lim_(n-gtoo) root(n)5(root(n)(11/5) - 1)/(1/n)`

`Use ` `(11/5)^(1/n) =root(n)(11/5).`

`lim_(n-gtoo) root(n)5((11/5)^(1/n) - 1)/(1/n)`

The new expression of the function resembles the basic fraction `(a^n - 1)/n`  whose limit is ln a.

`lim_(n-gtoo) root(n)5((11/5)^(1/n) - 1)/(1/n)=lim_(n-gtoo) root(n)5 *lim_(n-gtoo) ((11/5)^(1/n) - 1)/(1/n)=5^(1/oo)*ln (11/5) = 5^0*ln (11/5) = ln (11/5)`

When n goes to `oo` , the function converges to `ln (11/5).`