Convergence Tests

Question 1

For the given test, give the form of the series and list the condition of convergence, divergence, and comments. — DIVERGENCE TEST —

Answer 1

SERIES FORM: sum(a^k)

CONVERGENCE: does not apply

DIVERGENCE: lim a_k ≠ 0

COMMENTS: Cannot be used to prove convergence

Question 2

For the given test, give the form of the series and list the condition of convergence, divergence, and comments. — GEOMETRIC SERIES —

Answer 2

SERIES FORM: ∑ar^k

CONVERGENCE: |r| < 1

DIVERGENCE: |r| ≥ 1

COMMENTS: If |r| < 1, then ∑ar^k = a/(1-r)

Question 3

For the given test, give the form of the series and list the condition of convergence, divergence, and comments. — INTEGRAL TEST —

Answer 3

SERIES FORM: ∑a_k, where a_k=f(k) and f is continuous, positive, and decreasing

CONVERGENCE: ∫ f(x)dx < ∞

DIVERGENCE: ∫ f(x)dx does not exist

COMMENTS: The value of the integral is not the value of the series.

Question 4

For the given test, give the form of the series and list the condition of convergence, divergence, and comments. — P-SERIES —

Answer 4

SERIES FORM: ∑1/k^p

CONVERGENCE: p > 1

DIVERGENCE: p ≤ 1

COMMENTS: Useful for comparison tests

Question 5

For the given test, give the form of the series and list the condition of convergence, divergence, and comments. — RATIO TEST —

Answer 5

SERIES FORM: ∑a_k where a_k > 0

CONVERGENCE: lim (a_k+1)/(a_k) < 1

DIVERGENCE: lim (a_k+1)/(a_k) > 1

COMMENTS: Inconclusive if lim (a_k+1)/(a_k) = 1

Question 6

For the given test, give the form of the series and list the condition of convergence, divergence, and comments. — ROOT TEST —

Answer 6

SERIES FORM: ∑a_k where a_k ≥ 0

CONVERGENCE: lim ^k√(a_k) < 1

DIVERGENCE: lim ^k√(a_k​) > 1

COMMENTS: Inconclusive if lim ^k√(a_k​) = 1

Question 7

For the given test, give the form of the series and list the condition of convergence, divergence, and comments. — DIRECT COMPARISON TEST —

Answer 7

SERIES FORM: ∑a_k, where a_k > 0

CONVERGENCE: 0 < a_k ≤ b_k, and ∑b_k converges

DIVERGENCE: 0 < b_k ≤ a_k, and ∑b_k diverges

COMMENTS: ∑a_k is given; you supply ∑b_k

Question 8

For the given test, give the form of the series and list the condition of convergence, divergence, and comments. — LIMIT COMPARISON TEST —

Answer 8

SERIES FORM: ∑a_k, where a_k > 0, b_k > 0

CONVERGENCE: 0 < lim (a_k/b_k) < ∞, and ∑b_k converges

DIVERGENCE: lim (a_k/b_k) and ∑b_k diverges

COMMENTS: ∑a_k is given; you supply ∑b_k

Question 9

For the given test, give the form of the series and list the condition of convergence, divergence, and comments. — ALTERNATING SERIES TEST —

Answer 9

SERIES FORM: ∑(-1)^ka_k, where a_k > 0, 0 < a_k+1 ≤ a_k

CONVERGENCE: lim a_k = 0

DIVERGENCE: lim a_k ≠ 0

COMMENTS: Remainder R_n satisfies R_n < a_n+1