Comparison Test

Question 1

State the Direct Comparison Test (DCT) for convergence.

Answer 1

If 0 ≤ a_n ≤ b_n and Σ b_n converges, then Σ a_n must also converge.

Question 2

State the Direct Comparison Test (DCT) for divergence.

Answer 2

If 0 ≤ b_n ≤ a_n and Σ b_n diverges, then Σ a_n must also diverge.

Question 3

What is the limit requirement for the Limit Comparison Test (LCT)?

Answer 3

The limit as n→∞ of (a_n / b_n) must be a finite, positive number (L > 0).

Question 4

In LCT, if the limit is a finite positive number, what is the conclusion?

Answer 4

Both series Σ a_n and Σ b_n share the same behavior (both converge or both diverge).

Question 5

LCT: What can you conclude if the limit of a_n/b_n is 0 and Σ b_n converges?

Answer 5

Σ a_n also converges.

Question 6

LCT: What can you conclude if the limit of a_n/b_n is ∞ and Σ b_n diverges?

Answer 6

Σ a_n also diverges.

Question 7

Benchmark Selection: What series should you compare Σ 1 / (n^2 + 5) to?

Answer 7

Σ 1 / n^2 (a convergent p-series).

Question 8

Benchmark Selection: What series should you compare Σ 1 / (n - 1) to?

Answer 8

Σ 1 / n (the divergent harmonic series).

Question 9

DCT: Does Σ 1 / (n^2 + n) converge?

Answer 9

Yes. 1 / (n^2 + n) < 1 / n^2, and Σ 1 / n^2 converges.

Question 10

DCT: Does Σ (n + 1) / n^2 converge?

Answer 10

No. (n + 1) / n^2 > n / n^2 = 1/n, and Σ 1/n diverges.

Question 11

LCT: Determine the convergence of Σ (2n + 1) / (n^3 - n + 5).

Answer 11

Converges. Compare to Σ 2n / n^3 = Σ 2/n^2 using LCT.

Question 12

LCT: Determine the convergence of Σ 1 / (sqrt(n^2 + 1)).

Answer 12

Diverges. Compare to Σ 1/n using LCT; the limit is 1.

Question 13

DCT: Determine the convergence of Σ (sin^2 n) / n^2.

Answer 13

Converges. (sin^2 n) / n^2 ≤ 1 / n^2, and Σ 1 / n^2 converges.

Question 14

LCT: Determine the convergence of Σ (n^2 + 1) / (n^4 + n^2).

Answer 14

Converges. Compare to Σ n^2 / n^4 = Σ 1/n^2 using LCT.

Question 15

DCT: Does Σ 1 / (3^n + 1) converge?

Answer 15

Yes. 1 / (3^n + 1) < 1 / 3^n, which is a convergent geometric series (r=1/3).

Question 16

LCT: Determine the convergence of Σ 1 / (2^n - 1).

Answer 16

Converges. Compare to Σ 1 / 2^n using LCT; the limit is 1.

Question 17

DCT: Determine the convergence of Σ (ln n) / n.

Answer 17

Diverges. For n ≥ 3, (ln n) / n > 1/n, and Σ 1/n diverges.

Question 18

LCT: Determine the convergence of Σ (sqrt(n) + 1) / (n^2 + n).

Answer 18

Converges. Compare to Σ sqrt(n) / n^2 = Σ 1 / n^(1.5).

Question 19

DCT: Does Σ 1 / (n!) converge?

Answer 19

Yes. For n ≥ 4, 1/n! < 1/n^2 (or 1/2^n), and Σ 1/n^2 converges.

Question 20

LCT: Determine the convergence of Σ (n + 5) / (n^2 - 1).

Answer 20

Diverges. Compare to Σ n / n^2 = Σ 1/n.

Question 21

DCT: Determine the convergence of Σ 1 / (n^3 + n^2 + 1).

Answer 21

Converges. 1 / (n^3 + n^2 + 1) < 1/n^3, and Σ 1/n^3 converges.

Question 22

LCT: Determine the convergence of Σ (2^n + 1) / (3^n - 1).

Answer 22

Converges. Compare to Σ (2/3)^n using LCT; the limit is 1.

Question 23

DCT: Does Σ (3 + cos n) / n^2 converge?

Answer 23

Yes. (3 + cos n) / n^2 ≤ 4 / n^2, and Σ 4/n^2 converges.

Question 24

LCT: Determine the convergence of Σ 1 / (n^2 - n).

Answer 24

Converges. Compare to Σ 1/n^2; the limit is 1.

Question 25

LCT: Determine the convergence of Σ tan(1/n).

Answer 25

Diverges. Compare to Σ 1/n; the limit as n→∞ of [tan(1/n) / (1/n)] is 1.

Question 26

DCT: Determine the convergence of Σ 1 / (e^n + n).

Answer 26

Converges. 1 / (e^n + n) < 1/e^n, which is a convergent geometric series.

Question 27

LCT: Determine the convergence of Σ (n!) / (n+2)!.

Answer 27

Converges. Simplifies to Σ 1 / (n+2)(n+1); compare to Σ 1/n^2.

Question 28

LCT: Determine the convergence of Σ sqrt(n^2 + 1) / (n^3 + 1).

Answer 28

Converges. Compare to Σ n / n^3 = Σ 1/n^2.

Question 29

DCT: Does Σ 1 / (n^n) converge?

Answer 29

Yes. For n ≥ 2, 1/n^n ≤ 1/n^2, and Σ 1/n^2 converges.

Question 30

True or False: If Σ a_n diverges and a_n ≤ b_n, then Σ b_n must diverge.

Answer 30

True. This is the Direct Comparison Test for divergence.