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.

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,

- 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:

                                ∑  

Where,

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

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 ).

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:

- 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 }.

∠acb = 8°

step-by-step explanation:

∠xba lies on the circle and is half the measure of the intercepted arc ax

hence

measure of arc ax = 2 × 42° = 84°

∠acb is outside the circle and is

∠acb = (arcab - arcax ), hence

∠acb = 0.5 (100 - 84) = 0.5 × 16° = 8°

step-by-step explanation:

we are given a graph for system of inequalities.

in order to write the system of inequalities for the given graph, we need to find the inequality for each of the line separately.

let us find the slope of dotted line first.

taking two coordinates on the graph.

let us say (0,0) and (3,1).

let us find the slope of the line now.

slope of the dotted line would be =.

y-intercept of dotted line is y=0 only.

plugging slope m= and b=0 in slope-intercept form y=mx+b, we get

.

the line is shaded up of that dotted line.

therefore, for shading up we would take greater than inequality symbol > .

and because it's a dotted line we would take only > symbol in place of = sign in above equation of line.

therefore, inequality for dotted line would be

let us take two points (3,1) and (0,4) on the solid line.

slope of the solid line would be =.

for the solid line we have y-intercept b= 4.

plugging slope m= -1 and b=4 in slope-intercept form y=mx+b, we get

y = -1x + 4.

the line is shaded down of that solid line.

therefore, for shading down we would take less than inequality symbol < .

and because it's a solid line we would take ≤ symbol in place of = sign in above equation of line.

therefore, inequality for solid line would be