Infinite Sequences and Series (BC Only)

The test is inconclusive.

The limit does not exist.

The series converges.

The series diverges.

$(x'(t), y'(t))$

$(t \cdot x(t), t \cdot y(t))$

$\frac{dx}{dt}, \frac{dy}{dt}$

$\int x(t) dt, \int y(t) dt$

The series converges only at x = -3 and diverges otherwise.

The series converges for $x < 7$ and diverges for $x > 7$.

The series converges for all real numbers except x = -3 due to asymptotic behavior near that point.

The series converges when absolute value of $\frac{x-3}{4}$ is less than or equal to one-half ($\left| \frac{x-3}{4} \right| \leq \frac{1}{2}$).

$\int_{a}^{b}\left(\frac{y}{x}\right) dt$

$\int_{a}^{b}\sqrt{(x')^2 + (y')^2} dxdy$

$\int_{a}^{b}(x)dy = \int_{a}^{b}(x)\frac{dy}{dt} dt$

$\int_{a}^{b}(y)dx = \int_{a}^{b}(y)\frac{dx}{dt} dt$

Geometric Series

Telescoping Series

Alternating Series

P-Series

Converge if $\pi > v$

Converge if $v > \pi$

Diverge if $\pi < v$

Diverge if $v < \pi$

No effect because alternation doesn't change absolute convergence tests outcomes.

Results in divergence due to increased growth rate causing higher limit values.

Results in conditional but not absolute convergence owing to more rapid oscillations.

Depends on relative sizes α compared b+c̅ raises questions about rate changes versus base increase effects.

$p = 1$

$p > 1$

$p < -2$

$p = 0$

Convergence changes from divergent to convergent as exponential growth outpaces factorial growth.

The new series diverges faster because both numerator and denominator grow at faster rates individually.

Convergence properties remain unchanged because both series exhibit similar behavior for large values of n.

Convergence changes from convergent to divergent as factorial growth outpaces exponential growth.

$k > 0$

$1 < k \leq e$

$k > e$

$0 < k \leq 1$