Confirm that the integral test can be applied to the series. then use the integral test to determine the convergence or divergence of the series. 1 5 + 1 7 + 1 9 + 1 11 + 1 13

The question is:

Confirm that the integral test can be applied to the series. Then use the integral test to determine the convergence or divergence of the series: 1/5 + 1/7 + 1/9 + 1/11 + 1/13 + ...

The series is divergent.

Step-by-step explanation:

Given the series:

1/5 + 1/7 + 1/9 + 1/11 + 1/13 + ...

We can rewrite this series in summation notation as

Sum from 1 to infinity [1/(2n + 3)]

For this test to work,

1/(2n + 3) must be positive, decreasing, and continuous.

The summation is convergent if the integral from 1 to infinity [1/(2n + 3)] dx is convergent, and divergent if the integral from 1 to infinity [1/(2n + 3)] dx is divergent.

Now, let us consider the integral from 1 to infinity [1/(2x + 3)] dx

This can be written as

limit as y -> infty{integral from 1 to y [1/(2x + 3)] dx}

= (1/2)limit as y -> infty{ln|2x + 3| from 1 to y}

= (1/2)limit as y -> infty{ln|2y + 3| - ln(5)}

= infinity

Therefore, the series diverges.

something that characterises a right triangle is the fact that we may use pythagoras' theorem to find any one of its sides given that we have the length of the other two sides. here, we have the length of the hypotenuse, √(87) yd, and one other side, √(23) yd, thus we can easily find the length of the remaining side using pythagoras' theorem as defined below:

c^2 = a^2 + b^2, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

thus, given that c = √(87) and a = √(23), we may substitute these into the formula we defined above to get:

c^2 = a^2 + b^2

(√87)^2 = (√23)^2 + b^2 (substitute c = √87 and a = √23 into the equation)

87 = 23 + b^2 (evaluate (√87)^2 = 87 and (√23)^2 = 23 - remember that the square root of a number, squared will equal that very same number)

64 = b^2 (subtract 23 from both sides)

√64 = b (take the square root of both sides)

8 = b (evaluate √64 = 8)

therefor, the length of the unknown leg in the right triangle is 8 yd (first answer).

45

step-by-step explanation: