Integral Test and Improper Integrals

The video tutorial explains how to use the integral test to determine the convergence or divergence of a series. It begins by defining a function that is positive, decreasing, and continuous over a specified interval. The tutorial then demonstrates how to calculate the improper integral of this function, using the arctangent for the antiderivative. The result shows that if the integral converges, the series also converges. The video concludes with a summary of the process and highlights the importance of the integral test in analyzing series.

What is the first step in using the integral test to determine if a series converges or diverges?

• Use a calculator to approximate the series.

• Calculate the sum of the series.

• Graph the series.

• Find a function that is positive, decreasing, and continuous.

Which of the following is NOT a property that the function f(x) must have for the integral test?

• Decreasing

• Continuous

• Increasing

• Positive

What function is chosen for f(x) in the integral test example?

• x squared plus 1

• 1 divided by x

• 5 divided by the quantity 1 plus x squared

• 5 times x

Why is the integral from 1 to infinity considered improper?

• Because it is not decreasing.

• Because it has an infinite upper limit.

• Because it is not continuous.

• Because it is not positive.

What mathematical concept is used to evaluate the improper integral?

• Derivative

• Limit

• Matrix

• Factorial

Which function is used to find the anti-derivative in this example?

• Arctangent

• Exponential

• Sine

• Cosine

What value does arctangent approach as its input approaches infinity?

• Zero

• Pi over two

• Negative infinity

• One

What is the result of the integral from 1 to infinity of f(x) in this example?

• One

• Infinity

• Zero

• Five pi over four

What conclusion can be drawn if the integral converges?

• The series converges.

• The series is finite.

• The series diverges.

• The series is undefined.

What might be necessary if arctangent does not yield a nice value?

• Ignore the result.

• Change the function f(x).

• Recalculate the integral.

• Use a calculator for a decimal approximation.