Infinite Sequences and Series (BC Only)

A discontinuous function that is negative and increasing on [4, ∞)

A continuous function that oscillates between positive and negative values from [4, ∞)

A continuous, positive, decreasing function on [4, ∞)

A continuous function with no requirement on sign on [4, ∞)

Ratio Test, since $\frac{(-1)^n}{n!}$ expresses series with factorial terms indicating possible applicability.

Integral Test, since this test applies more directly to series resembling improper integrals.

p-Series Test, as this test applies specifically to series in form $\sum_{n=1}^{\infty} \frac{1}{n^p}$, but our given series contains factorial terminology.

Alternating Series Test, since alternating signs may suggest its use but won't confirm absolute convergence.

The terms of the sequence increase indefinitely.

The series converges absolutely.

Additional tests are needed to determine convergence.

The series diverges.

The series converges absolutely because the terms are bounded between -one and one.

It cannot be determined without applying a specific test for convergence.

The series converges conditionally but not absolutely.

The series diverges since $\cos(n)$ does not approach zero as n approaches infinity.

Ratio test

Root test

Integral test

Comparison test

Conditionally converge

Oscillate between finite bounds

Diverge

Absolutely converge

Find $\frac{dy}{dt}$ and $\frac{dx}{dt}$ and use the formula $\frac{dy/dt}{dx/dt}$.

Solve for $t$ in both parametric equations and then find the derivative.

Directly differentiate $x$ with respect to $y$.

Integrate $\frac{dy}{dt}$ with respect to $t$.

Root Test

Integral Test

Alternating Series Test

Ratio Test

Ratio test because factorials grow faster than exponential functions.

Root test because raising each term to the power $k$ may simplify comparison.

Integral test since factorials are associated with continuous growth functions.

Direct comparison test comparing each term with a simpler p-series or geometric series.

($\pi/2$, 4)

(4, $\pi/2$)

(-4, $\pi/2$)

(4, $\pi/3$)