Infinite Sequences and Series (BC Only)

(0, 2)

[-1, 1]

(-1, 1)

(\infty, -\infty)

Plot points for various values of $t$.

Integrate both functions with respect to $t$.

Eliminate the parameter $t$.

Find the derivatives $\frac{dx}{dt}$ and $\frac{dy}{dt}$.

Any real number value for $k$.

No value of $k$ makes it possible.

Only when $k=0$.

Only when $k=1$.

The geometric series $\frac{1000\pi}{1 - (0.5t)}$

The power series $4\pi(10)^2 \cdot (0.5t) + \frac{4\pi}{3}(0.5)^3t^3$

The power series $(4/3)\pi(10 + 0.5t)^3$

The Taylor series expansion at $r=10$ of $\pi r^2$

$( r\cos(\theta), r\sin(\theta) )

$(-r\sin(\theta), r\cos(\theta))$

$( e^{r\cos(\theta)}, e^{r\sin(\theta)} )

$( r + \cos(\theta), r + \sin(\theta) )

$\frac{dx}{dt} = \ln(3)e^{3t}$

$\frac{dx}{dt} = 3e^{3t}$

$\frac{dx}{dt} = te^{3t}$

$\frac{dx}{dt} = e^{3t}$

Root Test

Alternating Series Test

Integral Test

Ratio Test

Integrate each term from negative to positive infinity.

Divide each term by $x^n$ and sum up all terms.

Use the ratio test or root test on the terms of the series.

Multiply each term by $n^2$ and take limits.

Integral From G-H $[\cot(\degree)^3 d\theta]$

Integral form C-D $[\frac{2}{3} \sin(\degree) d\theta]$

Integral Form E-F $[\frac{1}{3} \tan(\degree) d\theta]$

Integral form A - B $[\frac{1}{2} r^2 d\theta]$

The radius is calculated by finding where the composed function becomes singular or discontinuous, which in this case requires a numerical or graphical approach to identify.

One calculates diverging points by setting inside function equal to plus/minus pi halves and then exponentially mapping back to the original variable $x$

If you look at it sideways, it's like calculating where $\tan^{-1}(\ln(x))$ equals zero and then determining convergence based on distances from those points.

The radius always equals to one since $\ln(x)$ approaches infinity as $x$ approaches $e$.