`sum_(n=1)^oo (-1)^n/sqrt(n)` Determine whether the series converges absolutely or conditionally, or diverges.

Expert Answers

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

To determine the convergence or divergence of the series `sum_(n=1)^oo (-1)^n/sqrt(n)` , we may apply the Root Test.

In Root test, we determine the limit as:

`lim_(n-gtoo) root(n)(|a_n|)= L`


`lim_(n-gtoo) |a_n|^(1/n)= L`

 Then ,we follow the conditions:

a) `L lt1` then the series converges absolutely

b) `Lgt1` then the series diverges

c) `L=1` or does not exist  then the test is inconclusive.The series may be divergent, conditionally convergent, or absolutely convergent.

For the given series `sum_(n=1)^oo (-1)^n/sqrt(n)` , we have `a_n =(-1)^n/sqrt(n).`

Applying the Root test, we set-up the limit as: 

`lim_(n-gtoo) |(-1)^n/sqrt(n)|^(1/n) =lim_(n-gtoo) (1/sqrt(n))^(1/n) Note: |(-1)^n| = 1`

Apply radical property: `root(n)(x) =x^(1/n) ` and Law of exponent: `(x/y)^n = x^n/y^n` .

`lim_(n-gtoo) (1/sqrt(n))^(1/n) =lim_(n-gtoo) (1/n^(1/2))^(1/n)`

                         ` =lim_(n-gtoo) 1^(1/n) /n^(1/2*1/n)`

                         ` =lim_(n-gtoo) 1^(1/n) /n^(1/(2n))`

                         ` =lim_(n-gtoo) 1 /n^(1/(2n))`

Apply the limit property: `lim_(x-gta)[(f(x))/(g(x))] =(lim_(x-gta) f(x))/(lim_(x-gta) g(x))` .

`lim_(n-gtoo) 1 /n^(1/(2n)) =(lim_(n-gtoo) 1 )/(lim_(n-gtoo)n^(1/(2n)))`

                    ` = 1/1`

                    ` =1`

The limit value `L = 1` implies that the series may be divergent, conditionally convergent, or absolutely convergent.

To verify, we use alternating series test on `sum a_n` .

`a_n = 1/sqrt(n)` is positive and decreasing from `N=1` 

`lim_(n-gtoo)1/sqrt(n) = 1/oo = 1`

Based on alternating series test condition,  the series  `sum_(n=1)^oo (-1)^n/sqrt(n)` converges.

Apply p-series test on `sum |a_n|` .

`sum_(n=1)^oo |(-1)^n/sqrt(n)|=sum_(n=1)^oo 1/sqrt(n).`

                     `=sum_(n=1)^oo 1/n^(1/2)`

Based on p-series test condition,  we have `p=1/2` that satisfies `0ltplt=1` .

Thus, the series  `sum_(n=1)^oo |(-1)^n/sqrt(n)|` diverges.                             


`sum_(n=1)^oo (-1)^n/sqrt(n)` is conditionally convergent since  `sum_(n=1)^oo (-1)^n/sqrt(n)` is convergent  and  `sum_(n=1)^oo |(-1)^n/sqrt(n)|` is divergent.

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial