Convergence Tests

Question 1

Divergence Test

Answer 1

lim n→ ∞ aₙ = L

1. if L ≠ 0 Σ aₙ diverges
2. if L = 0 test is inconclusive

Question 2

P-Series

Answer 2

aₙ = 1/(nᴾ), n ≥ 1
if p > 1 Σ aₙ converges
if p ≤ 1 Σ aₙ diverges

Question 3

Geometric Series

Answer 3

aₙ = arⁿ⁻¹, n ≥ 1
if |r| < 1 Σ (n = 1, ∞) aₙ = a/ (1-r)
if |r| ≥ 1 Σ aₙ diverges

Question 4

Alternating Series

Answer 4

aₙ = (-1)ⁿbₙ or aₙ = (-1)ⁿ⁺¹bₙ, b ≥ 0
Requirements:
1. bₙ₊₁ ≤ bₙ
2. lim n→ ∞ bₙ = 0 (Divergence Test)
Σ aₙ converges

Question 5

Telescoping Series

Answer 5

If subsequent terms cancel out previous terms in the sum. You may have to use partial fractions, properties of logarithms, etc. to put in appropriate form.
lim n→ ∞ sₙ = s
1. if s is finite Σ aₙ = s
2. if s isn’t finite Σ aₙ diverges

Question 6

Comparison Test

Answer 6

Pick {bₙ}

1. if Σ bₙ converges and 0 ≤ aₙ ≤ bₙ then Σ aₙ converges
2. if Σ bₙ diverges and 0 ≤ bₙ ≤ aₙ then Σ aₙ diverges

Question 7

Limit Comparison Test

Answer 7

Pick {bₙ}
1. lim n→ ∞ aₙ / bₙ = c where c > 0 and c is finite
2. aₙ, bₙ > 0
If Σ (n = 1, ∞) bₙ converges Σ aₙ converges
If Σ (n = 1, ∞) bₙ diverges Σ aₙ diverges

Question 8

Integral Test

Answer 8

aₙ = f(n)
Requirements for [a,∞):
1. f(x) is continuous
2. f(x) is positive
3. f(x) is decreasing
if ∫ (a,∞) f(x) converges Σ (n = a, ∞) aₙ converges
if ∫ (a,∞) f(x) diverges Σ aₙ diverges

Question 9

Ratio Test

Answer 9

lim n→ ∞ |aₙ₊₁/aₙ| = L

1. If L < 1 Σ aₙ absolutely converges
2. if L = 1 test is inconclusive
3. if L > 1 Σ aₙ diverges

Question 10

Root Test

Answer 10

lim n→ ∞ ⁿ√|aₙ| = L

1. If L < 1 Σ aₙ absolutely converges
2. if L = 1 test is inconclusive
3. if L > 1 Σ aₙ diverges

Question 11

If c is a real, positive number, that the limit of the sequence c¹/ ⁿ →

Answer 11

1

Question 12

If c is a real, positive number, then 1/nᶜ →

Answer 12

0

Question 13

cⁿ/n! →

Answer 13

0

Question 14

n¹/ ⁿ →

Answer 14

1

Question 15

(1 + c/n)ⁿ →

Answer 15

eᶜ

Question 16

tan⁻¹(x)

Answer 16

Σ((-1)ⁿx²ⁿ⁺¹)/(2n+1) from n=0 to ∞

[-1,1]

Question 17

ln(1+x)

Answer 17

Σ((-1)ⁿxⁿ⁺¹)/(n+1) from n=0 to ∞

(-1,1]

Question 18

cos(x)

Answer 18

Σ((-1)ⁿx²ⁿ)/(2n)! from n=0 to ∞

-∞,∞