All Flashcards
What is the general form of the Integral Test?
If is positive, continuous, and decreasing on and , then converges if and only if converges.
What is the integral test formula for convergence?
If converges, then converges.
What is the integral test formula for divergence?
If diverges, then diverges.
What is the integral of ?
What is the integral of ?
What is the formula for u-substitution?
, where and
What is the formula for the integral of a constant?
Give the formula for integration by substitution.
What is the formula for the arctangent integral?
What is the formula for the natural logarithm integral?
If the graph of is always above the x-axis and decreasing, what does this imply for the Integral Test?
It suggests that is positive and decreasing, which are two conditions needed to apply the Integral Test.
How can you visually determine if a function is decreasing from its graph?
The graph should be going downwards as you move from left to right.
If the area under the curve of from to is finite, what does this mean for the series ?
It suggests that the series converges, according to the Integral Test.
How does the graph of the derivative of help in the Integral Test?
If for , then is decreasing, a condition for the Integral Test.
How can the graph of be used to illustrate the Integral Test?
The graph shows a positive, decreasing function. The area under the curve from 1 to infinity can be compared to the sum of the series .
What does the area under the curve of represent in the context of the Integral Test?
It represents the value of the improper integral , which is compared to the sum of the series.
How does the steepness of the graph of relate to the convergence of the series?
A steeper decreasing graph suggests faster convergence, while a shallower decreasing graph may lead to divergence.
What does a graph that oscillates above and below the x-axis imply for the Integral Test?
The Integral Test cannot be applied since the function must be positive.
How does a discontinuity in the graph of affect the Integral Test?
If the discontinuity occurs within the interval of integration, the Integral Test cannot be directly applied.
How does the graph of illustrate the Integral Test?
The graph is positive and decreasing. The area under the curve from 1 to infinity converges, illustrating the convergence of the series .
How do you determine if converges using the Integral Test?
- Verify is positive, continuous, and decreasing for . 2. Evaluate . 3. If the integral converges, the series converges; if it diverges, the series diverges.
How do you evaluate ?
- Use u-substitution: , . 2. Rewrite the integral as . 3. Evaluate to get . 4. Evaluate the improper integral.
How do you show that is decreasing for ?
Show that the derivative is negative for .
Steps to apply the Integral Test.
- Verify is positive, continuous, and decreasing for . 2. Evaluate the improper integral . 3. Determine if the integral converges or diverges. 4. Conclude the series converges/diverges accordingly.
Given , how do you choose the u-substitution?
Choose because its derivative is also present in the series/integral.
How do you check if a function is decreasing?
Compute the derivative of the function. If the derivative is negative over the interval of interest, the function is decreasing.
How do you evaluate an improper integral with an infinite upper limit?
Replace the infinite limit with a variable, say , and take the limit as approaches infinity.
How do you determine if converges or diverges using the integral test?
Apply the integral test with . The series converges if and diverges if .
How to identify a suitable function for the Integral Test?
Look for a function such that matches the terms of the series and is positive, continuous, and decreasing.
How do you handle the lower limit of integration when using the Integral Test?
Ensure that the function satisfies the conditions of the Integral Test for all greater than or equal to the lower limit.