Infinite Sequences and Series (BC Only)

$k=0$

$k=1$

$k=2$

$k=-1$

$-\infty$

$1$

$+\infty$

$0$

It converges because it's an alternating series with decreasing magnitude.

It diverges because $\lim_{n \to \infty} a_n \neq 0$.

It converges absolutely because $\sin\left( \frac{\pi}{n} \right) \to 0$ as $n \to \infty$.

Nothing can be concluded because $\lim_{n \to \infty} a_n = 0$ does not ensure convergence.

P-series test

Convergence test

Ratio test

Divergence test

It approaches infinity.

It oscillates without approaching any value.

It approaches 0.

It approaches 1.

It indicates divergence since all terms increase without bound as $n$ grows large.

It proves convergence since all terms approach zero as $n$ increases indefinitely.

It shows divergence since $\lim_{n \to \infty} a_n = 0$, indicating no conclusion about convergence can be made from this test alone.

It suggests conditional convergence based on alternating term signs and diminishing value toward infinity.

Whether or not terms of p-series form an arithmetic progression

Whether or not terms of the p-series approach zero as n approaches infinity

If all terms of p-series are positive integers

If there exists an upper bound on terms of p-series as n increases indefinitely

It remains divergent.

It becomes conditionally convergent.

Convergence cannot be determined without additional information.

It becomes absolutely convergent.

The sequence converges to 0.

The series $\sum a_n$ diverges by the nth term test.

The nth term does not approach a limit as $n$ approaches $\infty$.

The sequence diverges to $\infty$.

It suggests it must converge due to presence of both negative and positive terms.

It diverges due to factorials increasing faster than any exponential function.

Little can be asserted about either convergence or divergence solely from examining $\lim_{n \to \infty} a_n = 0$.

Its rapid growth implies absolute convergence.