Infinite Sequences and Series (BC Only)

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 $e$.

Direct integration of $\frac{\sin(x)}{x^2}$ and then finding its Maclaurin series.

Maclaurin series expansion of $\sin(x)$ followed by term-by-term division.

Using geometric series representation for $\frac{1}{x^2}$ followed by multiplication with $\sin(x)$.

Substituting $x^2$ with a Taylor polynomial and then applying L'Hôpital's rule.

$1+5x+\frac{25}{2}x^2+\frac{125}{6}x^3$

$-1-5x-\frac{25}{2}x^2-\frac{125}{6}x^3$

$-1+5x-\frac{25}{2}x^2+\frac{125}{6}x^3$

$1-5x+\frac{25}{2}x^2-\frac{125}{6}x^3$

An alternating series that eventually leads to divergence regardless of sign changes

An alternating series whose absolute values decrease monotonically rapidly enough to ensure summability

Large numbers with significant digits are required to accurately compute the sums due to the delicate balance of subtraction involved

An alternating series exhibiting periodicity and predictable patterns at regular intervals

$1 - \frac{25}{2} x^2 + \ldots$

$25x + \frac{625}{6} x^3 + \ldots$

$-25x + \frac{625}{6} x^3 - \ldots$

$1 + \frac{25}{2} x^2 + \ldots$

A constant multiple of ${x^8}/5^{12/5}$ constitutes this term.

The ${x^8}$ term does not exist in this series.

The coefficient involves combinatorial expression $\binom{-4/5}{8}$.

It's given by multiplying ${-4/5 \choose {8}}$ by ${5^{-36/5}}$.

The Ratio Test

The Root Test

The Comparison Test

The Interval of Convergence Theorem

Minimum five terms

Minimum three terms

Minimum two terms

Minimum four terms

The set of all x-values for which the series converges.

The set of all x-values less than or equal to c where the series converges.

The set of all x-values where the integral of the series converges.

The set of all x-values for which the derivative of the series converges.

Differentiate the Taylor expansion of another function that converges to $\ln(x)$ at $x=1$.

Use integration by parts on the integral definition of $\ln(x)$ followed by a term-by-term integration.

Find the power series for $\ln(x)$ first, then differentiate term-by-term.

Apply L'Hospital's Rule to compute derivatives before forming the power series.