Infinite Sequences and Series (BC Only)

The series diverges because Leibniz's test dictates that alternating terms must exhibit symmetrical behavior in effect.

Leibniz's test remains unhelpful due to lacking ability to predict behavior based on non-constant amplitude variations.

Series convergence cannot be assured without complementary methodology given the uncertainty over term-to-term discrepancy.

That series converges since Leibniz's criteria for alternating series are satisfied and the term decreases to zero.

$\sum_{n=1}^{\infty} \frac{n^3}{n^{2}}$

$\sum_{n=1}^{\infty} \frac{5n}{n^{2}}$

$\sum_{n=1}^{\infty} \frac{\ln(6)}{n^{2}}$

$\sum_{n=1}^{\infty} \frac{5}{2n^{2}+4n+3}$

Neither of the series converge.

Only the second series converges by the geometric series test.

Both series converge by p-series and geometric tests.

Only the first series converges by the p-series test.

$p < 2$

$p \leq 1$

$p > 0$

$p = \frac{3}{2}$

The first model with $\sum_{n=0}^{\infty} \frac{5^n}{(2n)!}$

Both models converge at the same rate.

Neither model will provide quick stabilization due to slow convergence.

The second model with $\sum_{n=0}^{\infty} \frac{3^n}{(3n)!}$

It diverges by comparison to a p-series because $-\pi < -1$ indicates divergence of terms involving powers of $-\pi$.

It converges by comparison to a p-series because $n^{-\pi}$ satisfies $p > 1$.

It diverges by comparison to a geometric series since $\left|\cos\left(\frac{\pi}{4}\right)\right| < 1$ and acts as a common ratio.

It converges by comparison to a geometric series since $\left|\cos\left(\frac{\pi}{4}\right)\right| < 1$ and acts as a common ratio.

Direct Comparison Test comparing $\frac{\sin(x)}{x^{3/2}} dx$ with $\frac{dx}{x^{\nu^{\pu}}}$

Absolute Convergence Test

Limit Comparison Test

Rational Root Test

$p < +1$

$p > 1$

$p = +1$

$p < -1$

Absolute convergence implies limited energy ranges whereas conditional suggests broader predictability across all energies.

Neither type offers reliable predictions unless supplemented with experimental data.

Both types of convergence result in identical predictions for particle behavior across varied energies.

Conditional convergence implies limited energy ranges whereas absolute suggests broader predictability across all energies.

It changes from converging to diverging.

It changes from diverging to converging.

No effect; it remains diverging.

No effect; still ultimately converges.