Test the series for convergence or divergence. [infinity] (−1)n + 1 3n5 n = 1 converges diverges Correct: Your answer is correct. If the series is convergent, use the Alternating Series Estimation Theorem to determine how many terms we need to add in order to find the sum with an error less than 0.00005. (If the quantity diverges, enter DIVERGES.) terms

a.

The sequence that you are summing is decreasing and its limit is zero, therefore it converges.

b.

You need to sum at least 6 terms to get that precision

Step-by-step explanation:

You want to determine the convergence of the series

Since.

1.   is a decreasing sequence

2.

Therefore by the theorem of convergence for alternating series i is convergent.

To determine how many terms you need to add in order to find the sum with an error less than 0.00005 we use the estimation theorem for alternating series which states that for the n-error of an alternating series is upper bounded by the next term of the series. In math terms it looks like this.

Given

The error satisfies that

For this problem, we are looking for an such that

then, solving the inequality you get that

Therefore you need to sum at least 6 terms to get that precision

p ÷ r = tan(x)

step-by-step explanation:

given that there is a figure below shows a right triangle: a right triangle is shown with hypotenuse equal to q units and the length of the legs equal to p units and r units.

graph is missing so i will create the graph as per given information.

the angle between the legs having lengths p units and q units is y degrees. the angle between the legs having lengths r units and q units is x degrees.

now we need to find about what is p ÷ r equal to.

so we can use trigonometric ratio tan(x)=[altitude]/[base] to find the value of p ÷ r, because p is the altitude and r is the base.

p ÷ r=tan(x)

hence final answer is p ÷ r=tan(x).