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 $e$.

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

If only every term in the original series contains positive exponents at that point.

If both the original series and its term-by-term derivative converge at that point.

If there are no coefficients equal to zero in any terms around that point.

If only its term-by-term integral converges at that point.

Employ substitution in Cauchy-Hadamard formula post-series formation.

Directly use the Radius of Convergence from the original exponential function without substitution.

Substitute $x^2$ into the expansion of $e^u$, where $u=x^2$, and then apply the Ratio Test.

Utilize comparison with geometric series after identifying leading coefficients in each term.

The terms must increase monotonically and remain positive.

The series must have a fixed number of terms.

The series must converge conditionally.

The terms $f(n)$ must decrease monotonically toward zero and the limit as $n$ approaches infinity must be zero.

$( \frac{\pi}{2}, 5 )

$( 5, \frac{\pi}{4} )

$(-5, \frac{\pi}{2})

$( 5, \frac{\pi}{2} )

Increasing Sequence

Cyclic Sequence

Bounded Sequence

Decreasing Sequence

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

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

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

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

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 ${5^{-36/5}}$.

Minimum five terms

Minimum three terms

Minimum two terms

Minimum four terms