All flashcards
Flashcard 1: What is the outcome if the series converges in the Integral Test?
Answer: The integral also converges. The test establishes this direct correspondence.
Flashcard 2: What is the result of a converging integral in the Integral Test?
Answer: The series converges. Convergent integral guarantees convergent series.
Flashcard 3: How does the Integral Test determine convergence?
Answer: By comparing the series to the integral of the corresponding function. The test links series behavior to integral behavior.
Flashcard 4: What ensures the series' convergence in the Integral Test?
Answer: The integral must converge. Convergent integral is the key condition.
Flashcard 5: State a condition invalidating the Integral Test.
Answer: If f(x) is not positive, continuous, or decreasing. All requirements must be met for validity.
Flashcard 6: What is the Integral Test's implication of a diverging integral?
Answer: The series diverges. The behaviors are equivalent under the test.
Flashcard 7: Which condition invalidates the Integral Test?
Answer: If the function is not continuous, positive, or decreasing. Any missing condition prevents using the test.
Flashcard 8: State the result of a diverging integral in the Integral Test.
Answer: The series diverges. Integral and series have equivalent behavior.
Flashcard 9: What confirms a series' divergence in the Integral Test?
Answer: A diverging integral. Divergent integral signals divergent series.
Flashcard 10: What is the implication of a converging integral in the Integral Test?
Answer: The series converges. The test creates this direct equivalence.
Flashcard 11: Identify the behavior of f(x) required for the Integral Test.
Answer: f(x) must be positive, continuous, and decreasing. All three properties must be satisfied.
Flashcard 12: Identify the type of series applicable for the Integral Test.
Answer: Series with positive, continuous, and decreasing terms. The test only works with these specific properties.
Flashcard 13: Which type of function is necessary for the Integral Test?
Answer: A positive, continuous, decreasing function. All three properties are essential for the test.
Flashcard 14: What does the Integral Test verify?
Answer: It verifies the convergence of series through integrals. The test establishes series convergence through integrals.
Flashcard 15: What must be true for f(x) for the Integral Test?
Answer: f(x) must be positive, continuous, and decreasing. All three properties are necessary conditions.
Flashcard 16: What must be true for f(x) to use the Integral Test?
Answer: f(x) must be positive, continuous, and decreasing. These conditions are mandatory for the test.
Flashcard 17: What is the outcome if the series converges in the Integral Test?
Answer: The integral also converges. The test establishes this direct correspondence.
Flashcard 18: What is required for f(x) for the Integral Test?
Answer: f(x) must be positive, continuous, and decreasing. These are the necessary function conditions.
Flashcard 19: What does the Integral Test require about the series terms?
Answer: The terms must be positive, continuous, and decreasing. These properties enable the integral comparison.
Flashcard 20: What characterizes the function in the Integral Test?
Answer: f(x) must be positive, continuous, and decreasing. All three properties define the applicable function.
Flashcard 21: What indicates a converging series in the Integral Test?
Answer: A converging integral. Convergent integral signals convergent series.
Flashcard 22: What does the Integral Test require about the function f(x)?
Answer: f(x) must be positive, continuous, and decreasing. These are the three fundamental requirements.
Flashcard 23: State the necessity of a positive function in the Integral Test.
Answer: The function must be positive for the Integral Test to apply. Positive values are required for meaningful comparison.
Flashcard 24: What is the consequence of a diverging integral in the Integral Test?
Answer: The series diverges. Divergent behaviors are equivalent under the test.
Flashcard 25: What determines the applicability of the Integral Test?
Answer: The function must be positive, continuous, and decreasing. These conditions ensure the test works properly.
Flashcard 26: What ensures the Integral Test is applicable to a series?
Answer: The series terms must form a positive, continuous, decreasing function. All conditions must hold for test validity.
Flashcard 27: What is the Integral Test's requirement for f(x)?
Answer: f(x) must be positive, continuous, and decreasing. These are the essential function properties.
Flashcard 28: What is the implication of a converging integral for the series?
Answer: The series converges. Convergent integral guarantees convergent series.
Flashcard 29: Define the convergence criterion in the Integral Test.
Answer: The integral must converge for the series to converge. Integral convergence is equivalent to series convergence.
Flashcard 30: What are the requirements for f(x) in the Integral Test?
Answer: f(x) must be positive, continuous, and decreasing. All three conditions are essential.