Infinite Sequences and Series (BC Only)

Ratio Test

Alternating Series Test

Integral Test

Root Test

5+cos(θ)=6

r=5/θ

5sin(θ)=6

r=5

The value of $\frac{df}{dt}$

The value of $(\int f(t) dt, \int g(t) dt)$

The values $(f(t), g(t))$

The value of $(\frac{d^2f}{dt^2}, \frac{d^2g}{dt^2})$

Ratio test

p-series test

Integral test

Root test

It diverges because it behaves like a p-series with p ≤ 1.

It conditionally converges by comparing it to an alternating harmonic series.

It diverges because it's greater than a known divergent harmonic series.

It converges absolutely because it's less than a convergent geometric series.

The Ratio Test, applying the limit as $n$ approaches infinity of absolute values.

The Root Test by examining the limit of the $n$-th root of terms.

The Limit Comparison Test with the convergent series $\sum \frac{1}{n}$.

Direct Comparison Test with the divergent series $\sum \frac{1}{n^{3/2}}$.

The Ratio Test confirms convergence because limit approaches zero as k goes to infinity.

Applying D'Alembert's ratio test yields no useful information for factorial terms.

The Ratio Test is inconclusive because limit approaches one as k goes to infinity.

It indicates divergence since limit exceeds one as k goes to infinity.

Comparison with $\sum_{n=5}^{\infty} \frac{n}{e^n}$

Comparison with $\sum_{n=5}^{\infty} \frac{2}{\sqrt{n}}$

Comparison with $\sum_{n=5}^{\infty} \frac{n}{\ln(n)}$

Comparison with $\sum_{n=5}^{\infty} \frac{1}{n^{2}}$

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, ∞)

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 $t$.