Question

When do you use the divergence test, comparison test, p test, geometric, integral, comparison, limit comparison, and alternating series?

Answer 1

Step-by-step explanation:

Divergence Test: use this when the limit as n approaches infinity of a sequence isn't 0.

If lim(n→∞) an ≠ 0, then an diverges.

(Note this only tests divergence, not convergence.)

P Test: use this when the series is a p-series.

For an = 1 / nᵖ, if p > 1, then the series converges.  Otherwise, it diverges.

Geometric Test: use this when the series is a geometric series.

For an = a₁ (r)ⁿ, if -1 < r < 1, then the series converges.  Otherwise, it diverges.

Integral Test: use this when the sequence can be easily integrated.

If ∫₁°° f(x) dx converges, then ∑₁°° f(n) converges.

If ∫₁°° f(x) dx diverges, then ∑₁°° f(n) diverges.

Comparison Test: use this when a sequence is similar to a p-series or a geometric series.

If bn > an and bn converges, then an converges.

If bn < an and bn diverges, then an diverges.

Otherwise, inconclusive.

Limit Comparison Test: use this when comparison test is inconclusive.

If an ≥ 0 and bn > 0, and lim(n→∞) an/bn > 0 and finite, then an and bn either both converge or both diverge.

Alternating Series Test: use this when the series is alternating.  This usually includes (-1)ⁿ or (-1)ⁿ⁺¹, but might use trig functions instead.

If an = (-1)ⁿ bn or (-1)ⁿ⁺¹ bn, where bn ≥ 0, and if lim(n→∞) bn = 0, and bn is decreasing, then an converges.

(Notice this only tests for convergence, not divergence.)