Infinite Sequences and Series (BC Only)

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

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

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

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

$k > 0$

$1 < k \leq e$

$k > e$

$0 < k \leq 1$

$L = \lim_{n \to \infty} \frac{(n+1)!/5^{n+1}}{n!/5^n}$

$L = \lim_{n \to \infty} \frac{n!/5^n}{(n-1)!/5^{n-1}}$

$L = 0$

$L = \lim_{n \to \infty} n!\cdot5^{-n}$

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 ($\left| \frac{x-3}{4} \right| \leq \frac{1}{2}$).

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

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

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

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

The first derivative has an additional factor of $e^{-6}$.

The first derivative becomes three times greater than that of $f(x)$.

The first derivative is unchanged except for a horizontal shift 2 units to the right.

The first derivative is multiplied by -1 compared to that of $f(x)$.

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.

Geometric Series

Telescoping Series

Alternating Series

P-Series

$p = 1$

$p > 1$

$p < -2$

$p = 0$

Converge if $\pi > v$

Converge if $v > \pi$

Diverge if $\pi < v$

Diverge if $v < \pi$