Calculus 2 Unit 9 Convergence Tests

Ratio Test for Absolute Convergence

Let Σu_k be a series with nonzero terms and suppose that p = lim k → ∞ |u_k + 1| / |u_k|

a) if p < 1 the series Σu_k converges absolutely

b) if p > 1 or p = ±∞ the series Σu_k diverges

c) if p = 1, no conclusion

Absolute Convergence Test

A series Σa_k converges absolutely if the series of abolsute values |Σa_k| converges. If |Σa_k| converges, then so does Σa_k.

Alternating Series Test

An alternating series of either form converges if:

1) a₁ > a₂ > a₃ > a₄ > … (eventually)

2) p = lim k → ∞ (a_k) = 0

Limit Comparison Test

Let Σa_k and Σb_k be a series with positive terms and consider p = lim k → ∞ a_k / b_k

If p is finite and p > 0 then both series either converge or diverge.

Ratio Test

Let Σa_k be a series with positive terms and consider p = lim k → ∞ (a_k+1) / (a_k)

a) If p < 1, the serires converges

b) if p > 1 or p = ±∞ the series Σu_k diverges

c) if p = 1, no conclusion

Root Test

Let Σa_k be a series with positive terms and suppose that p = lim k → ∞ ^k√(a_k) = lim k → ∞ (a_k)^1/k

a) If p < 1, the series converges

b) if p > 1 or p = ±∞ the series Σu_k diverges

c) if p = 1, no conclusion

Comparison Test

Let Σa_k and Σb_k be series of maginatude terms and suppose that a₁ ≤ b₁, a₂ ≤ b₂, a₃ ≤ b₃, …, a_n ≤ b_n, … (eventually)

a) If the “bigger” series Σb_k converges then Σa_k also converges

b) If Σa_k diverges, then Σb_k diverges

P-Series

The p-series Σ1/n^p converges if p > 1 and diverges if 0 < p ≤ 1

Integral Test

Let Σa_k be a series with positive terms. If f(x) is a function that is decreasing and continuous on the inverval [b, ∞) and a_k = f(k) ∀k ≥ b then Σa_k and ∫ ∞ b f(x)dx both either converge or diverge.

Divergence Test

If lim k → ∞ a_k ≠ 0 then the series Σa_k diverges.

If lim k → ∞ a_k = 0 then the series converges or diverges (inconclusive).

Geometric Series

a + ar + ar² + ar³ + … + ar^k + …

(a ≠ 0)

r is a ratio of series

It converges if |r| < 1 and diverges if |r| ≥ 0.

If the series converges then the sum if 1 / (1-r)

Harmonic Series

Σ1/n = 1 + ½ + 1/3 + … + 1/k + …

Diverges