Question

Determine whether the alternating series E (-1)^n+1 (n/8)^n converges or diverges. Choose the correct answer below​ and, if​ necessary, fill in the answer box to complete your choice. A. The series does not satisfy the conditions of the Alternating Series Test but diverges because it is a​ p-series with pequals nothing. B. The series does not satisfy the conditions of the Alternating Series Test but converges because it is a​ p-series with pequals nothing. C. The series does not satisfy the conditions of the Alternating Series Test but diverges by the Root Test because the limit used does not exist. D. The series does not satisfy the conditions of the Alternating Series Test but converges because it is a geometric series with requals nothing. E. The series converges by the Alternating Series Test.

Answer 1

Answer:

C

Step-by-step explanation:

Solution:-

- The Alternate series test is applicable for alternating series with has terms summed and subtracted alternatively and takes the form of:

       

                                   ∑ an

Were,

                                $a_n = ( -1 ) ^(^n^+^1^) b_n$

- Where, {  bn } > 0 for all n. Then if the following conditions are met:

1. Lim ( n -> ∞ ) { b_n } = 0

2. b ( n + 1 )  < bn  .... bn is a decreasing function.

Conclusion:- The series { ∑ an } is convergent.

- The following series is given as follows:

                                ∑  $( - 1 )^(^n^+^1^) (\frac{n}{8} )^n$

Where,

                               $b_n = (\frac{n}{8} )^n$

1 . We will first test whether the sequence { bn } is decreasing or not. Hence,

                              $b_n_+_1 - b_n < 0\\\\(\frac{n+1}{8})^(^n^+^1^) - (\frac{n}{8})^n\\\\(\frac{n}{8})^n ( \frac{n-7}{8} )$

We see that for n = 1 , 2 , 3 ... 6 the sequence { b_n } is decreasing; however, for n ≥ 7 the series increases. The condition is not met for all values of ( n ). Hence, the Alternating series test conditions are not satisfied.

We will now apply the root test that states that a series given in the following format:

                               ∑ an

- The limit of the following sequence { an } is a constant ( C ).

                               $C = Lim ( n - > inf ) [ a_n ] ^\frac{1}{n}$

1. C < 1 , The series converges

2.C > 1 , The series diverges

3. C = 1 , test is inconclusive

- We will compute the limit specified by the test as follows:

                          $Lim ( n - >inf ) = [ (\frac{n}{8})^n ]^\frac{1}{n} \\\\Lim ( n - >inf ) = [ (\frac{n}{8}) ] = inf$

- Here, the value of C = +∞ > 1. As per the Root test limit conditions we see that the series { ∑ an } diverges.

Note: Failing the conditions of Alternating Series test does not necessarily means the series diverges. As the test only implies the conditions of "convergence" and is quiet of about "divergence". Hence, we usually resort to other tests like { Ratio, Root or p-series tests for the complete picture }.