Unit 9 Convergence Tests

Question 1

Order of Tests

Answer 1

1. Determine if Absolute Value Test Can Be Used
2. Direct Recognition: Geometric, Telescoping, Harmonic, P-Series, Alternating, Alternating Harmonic
3. Divergence Test
4. Comparison Test
5. Limit Comparison Test
6. Integral Test
7. Ratio Test
8. Power Series

Question 2

Determine if Absolute Value Test Can Be Used

Answer 2

If Σ_n=1^∞ |a_n|converges, then Σ_n=1^∞ a_n converges.

ONLY WORKS IF Σ_n=1^∞ |a_n| CONVERGES

Use Comparison Test to prove Σ_n=1^∞ |a_n|converges.

Question 3

Geometric Series

Answer 3

• Direct Recognition
  -Looks like: a+ar+ar²+ar³+…
  -Converges to a/(1-r) if |r|<1 OR ELSE IT DIVERGES

Question 4

Telescoping Series

Answer 4

• Direct Recognition
  -Can Converge or Diverge
• Subsequent terms cancel out previous terms
• Use Partial Fractions or Law of Logs to put into proper form

Question 5

Harmonic Series

Answer 5

• Direct Recognition
• Diverges
• Grows without an upper bound
• EX: 1 + 1/2 + 1/3 + 1/4 + 1/5 + … + 1/n

Question 6

P-Series

Answer 6

• Direct Recognition
• Converges if p>1, Diverges if p≤1
• Form: Σ ¹⁄_x^p

Question 7

Alternating Series

Answer 7

• Direct Recognition
• Converges if lim_n→∞ b_n = 0 AND function is decreasing
• Form: Σ (-1)ⁿb_n)

Question 8

Alternating Harmonic Series

Answer 8

• Direct Recognition
• Converges
• Form: Σ ^(-1)n⁄_n

Question 9

Divergence Test

Answer 9

Diverges if lim_n→∞ a_n<= 0
Otherwise, it doesn’t tell you ANYTHING

Question 10

Comparison Test

Answer 10

Pick a function (b_n)

If b_n converges and 0 ≤ a_n ≤ b_n OG series converges.

If b_n diverges and 0 ≤ b_n ≤ a_n OG series diverges.

Question 11

Limit Comparison Test

Answer 11

Pick a function (b_n)

lim_n→∞ ^a_n⁄_bn > 0 and finite AND b_n > 0 AND a_n > 0

If Σ_n=1,∞ b_n converges, then OG series converges.

If Σ_n=1,∞ b_n diverges, then OG series diverges.

Question 12

Integral Test

Answer 12

If function f(x) a_n is continuous, positive, and decreasing, then:

If ∫_a^∞ f(x)dx converges, series converges

If ∫_a^∞ f(x)dx diverges, series diverges

Question 13

Ratio Test

Answer 13

If lim_n→∞ ^a_n+1⁄_an ≠ 1, then:

If lim_n→∞ |^a_n+1⁄_an| < 1, series absolutely converges

If lim_n→∞ |^a_n+1⁄_an| > 1, series absolutely diverges

Question 14

Power Series

Answer 14

For any Power Series Σ C_n(x-a)ⁿ, it can be one of the following:

1. Series only converges at x=a, otherwise it diverges. Radius of Convergence (R) = 0
2. Series converges absolutely (R=∞)
3. Series converges for Interval I: (a-R, a+R) and diverges outside

To find Interval of Convergence, use Ratio Test

Let a_n = C_n(x-a)ⁿ

1. If lim_n→∞ |^a_n+1⁄_an| = ∞, then series diverges everywhere but x=a (R=0)
2. If lim_n→∞ |^a_n+1⁄_an| = 0, then series converges everywhere (R=∞)
3. If lim_n→∞ |^a_n+1⁄_an| = K(x-a) where K is finite and non-zero, then it converges when K(x-a) < 1 ∴ |x-a| < ¹⁄_K ⇒ R=¹⁄_K and Interval: a-¹⁄_K<x<a+¹⁄_K