Infinite Sequences and Series (BC Only)

The sequence converges even though it diminishes slowly due to cancellation.

The sequence converges to $e$ because $\sqrt[k]{k}$ approaches $e$ asymptotically.

The sequence is bounded above by $e$ and below by $1$, but diverges because it increases without bound.

The sequence is simultaneously bounded above and below but diverges due to its oscillatory nature.

$-1 < p < 0$

$p > 1$

$p < -1$

$0 < p < 1$

The p-series test.

The root test.

The ratio test.

The alternating series test.

The series converges by oscillation as the terms alternate in sign and decrease in absolute value.

The series converges because the limit of the ratio of successive terms is zero.

The series diverges because the limit of the ratio of successive terms is greater than one.

Convergence cannot be determined with the Ratio Test due to indeterminate higher-order terms.

Integral Test

Ratio Test

Root Test

Direct Comparison Test

Integral Test

Root Test

Alternating Series Test

Ratio Test

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

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

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

$\sum_{n=1}^{\infty} \frac{7n^{2} + 4}{n}$

Integral Test

Root Test

Alternating Series Test

Ratio Test

Divergence Test

p-Series Test

Ratio Test

Root Test

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.