Integral Test for Convergence - AP Calculus BC

The series converges. Convergent integral guarantees convergent series.

By comparing the series to the integral of the corresponding function. The test links series behavior to integral behavior.

The integral must converge. Convergent integral is the key condition.

If $f(x)$ is not positive, continuous, or decreasing. All requirements must be met for validity.

The series diverges. The behaviors are equivalent under the test.

If the function is not continuous, positive, or decreasing. Any missing condition prevents using the test.

The series diverges. Integral and series have equivalent behavior.

A diverging integral. Divergent integral signals divergent series.

The series converges. The test creates this direct equivalence.

$f(x)$ must be positive, continuous, and decreasing. All three properties must be satisfied.

Series with positive, continuous, and decreasing terms. The test only works with these specific properties.

A positive, continuous, decreasing function. All three properties are essential for the test.

It verifies the convergence of series through integrals. The test establishes series convergence through integrals.

$f(x)$ must be positive, continuous, and decreasing. All three properties are necessary conditions.

$f(x)$ must be positive, continuous, and decreasing. These conditions are mandatory for the test.

The integral also converges. The test establishes this direct correspondence.

$f(x)$ must be positive, continuous, and decreasing. These are the necessary function conditions.

The terms must be positive, continuous, and decreasing. These properties enable the integral comparison.

$f(x)$ must be positive, continuous, and decreasing. All three properties define the applicable function.

A converging integral. Convergent integral signals convergent series.

$f(x)$ must be positive, continuous, and decreasing. These are the three fundamental requirements.

The function must be positive for the Integral Test to apply. Positive values are required for meaningful comparison.

The series diverges. Divergent behaviors are equivalent under the test.

The function must be positive, continuous, and decreasing. These conditions ensure the test works properly.

The series terms must form a positive, continuous, decreasing function. All conditions must hold for test validity.

$f(x)$ must be positive, continuous, and decreasing. These are the essential function properties.

The series converges. Convergent integral guarantees convergent series.

The integral must converge for the series to converge. Integral convergence is equivalent to series convergence.

$f(x)$ must be positive, continuous, and decreasing. All three conditions are essential.